3,240 results on '"HYPERGEOMETRIC functions"'
Search Results
2. Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type.
- Author
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Bychkov, Boris, Dunin‐Barkowski, Petr, Kazarian, Maxim, and Shadrin, Sergey
- Subjects
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PARTITION functions , *QUADRATIC equations , *HYPERGEOMETRIC functions - Abstract
We study the n$n$‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2$\hbar ^2$‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ℏ2$\hbar ^2$‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Relations of multiple t-values of general level.
- Author
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Li, Zhonghua and Wang, Zhenlu
- Subjects
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ZETA functions , *GENERATING functions , *HYPERGEOMETRIC functions - Abstract
We study the relations of multiple t -values of general level. The generating function of sums of multiple t -(star) values of level N with fixed weight, depth and height is represented by the generalized hypergeometric function 3 F 2 , which generalizes the results for multiple zeta(-star) values and multiple t -(star) values. As applications, we obtain formulas for the generating functions of sums of multiple t -(star) values of level N with height one and maximal height and a weighted sum formula for sums of multiple t -(star) values of level N with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple t -(star) values of level N. Some evaluations of multiple t -star values of level 2 with one–two–three indices are given. [ABSTRACT FROM AUTHOR]
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- 2024
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4. The Rheological Models of Becker, Scott Blair, Kolsky, Lomnitz and Jeffreys Revisited, and Implications for Wave Attenuation and Velocity Dispersion.
- Author
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Carcione, José M., Mainardi, Francesco, Qadrouh, Ayman N., Alajmi, Mamdoh, and Ba, Jing
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PHASE velocity , *HYPERGEOMETRIC functions , *MATHEMATICAL models , *EARTHQUAKES , *NUMERICAL calculations , *QUALITY factor , *RAYLEIGH waves , *CREEP (Materials) - Abstract
The rheological models of Lomnitz and Jeffreys have been widely used in earthquake seismology (to simulate a nearly constant Q medium) and to describe the creep and relaxation behavior of rocks as a function of time. Other similar models, such as those of Becker, Scott Blair and Kolsky, show similar properties, particularly the Scott Blair model describes a perfectly constant Q as a function of frequency. We first give a historical overview of the main scientists and the development and versions of the various models and priorities of discovery. Then, we clarify the relationship between the different versions of these models in terms of mathematical expressions of the complex modulus and calculate the phase velocity and quality factor Q as a function of frequency, illustrating the various special cases. In addition, we give useful hints for the numerical calculation of these moduli, which include special cases of the hypergeometric function. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Solving second‐order differential equations in terms of confluent Heun's functions.
- Author
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Aldossari, Shayea
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DIFFERENTIAL equations , *LINEAR differential equations , *HYPERGEOMETRIC functions - Abstract
This paper presents an algorithm that checks if a given second‐order homogeneous linear differential equation can be reduced to the confluent Heun's equation by using the change of variables transformation and the exp‐product transformation. The main purpose of this paper is finding solutions in terms of the confluent Heun's functions for the given differential equation. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices.
- Author
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Andrade, Julio C and Best, Christopher G
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POLYNOMIALS , *SYMPLECTIC groups , *UNITARY groups , *HYPERGEOMETRIC functions , *ZETA functions , *RANDOM matrices - Abstract
We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group S p (2 N) and the orthogonal ensembles S O (2 N) and O − (2 N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L -functions with symplectic and orthogonal symmetry. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Summing Sneddon-Bessel series explicitly.
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Durán, Antonio J., Pérez, Mario, and Varona, Juan Luis
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HYPERGEOMETRIC functions , *BESSEL functions , *HYPERGEOMETRIC series , *INTEGERS - Abstract
We sum in a closed form the Sneddon-Bessel series ∑∞ m=1 Jα(xjm,v)Jβ(yjm,v) /j m,v2n+α+β-2v+2 Jv+1(jm,v)², where 0 < x, 0 < y, x + y < 2, n is an integer, α, β, v ε C\{-1,-2, ... } with 2 Re v < 2n + 1 + Re α + Re β and {jm,v}m≥0 are the zeros of the Bessel function Jv of order v. In most cases, the explicit expressions for these sums involve hypergeometric functions pFq. As an application, we prove some extensions of the Kneser-Sommerfeld expansion. For instance, we show that ∑∞ m=1 j v-β m,v Jv (xjm,v)Jβ (yjm,v) (j²m,v-z²)Jv+1(jm,v)² = πJβ (yz)/4zβ-v Jv (z) (Yv (z)Jv (xz) - Jv (z)Yv (xz)), if Re v < Re β + 1 and 0 < y ≤ x, x + y < 2 (here, Yv denotes the Bessel function of the second kind), which becomes the Kneser-Sommerfeld expansion when β = v. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Some Properties of Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine.
- Author
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Zhang, Tao, Yang, Zhen-Hang, Qi, Feng, and Du, Wei-Shih
- Subjects
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SINE function , *HYPERGEOMETRIC functions , *INTEGRAL representations , *POWER series , *FRACTIONAL integrals , *COSINE function , *OPTIMISM - Abstract
In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Algebraic independence and linear difference equations.
- Author
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Adamczewski, Boris, Dreyfus, Thomas, Hardouin, Charlotte, and Wibmer, Michael
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LINEAR differential equations , *AUTOMORPHISMS , *ALGEBRAIC independence , *HYPERGEOMETRIC functions , *GALOIS theory - Abstract
We consider pairs of automorphisms acting on fields of Laurent or Puiseux series: pairs of shift operators .W x 7 x C h1; W x 7 x C h2/, of q-difference operators .W x 7 q1x, W x 7 q2x/, and of Mahler operators .W x 7 xp1 ; W x xp2 /. Given a solution f to a linear -equation and a solution g to an algebraic -equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the -Galois theory of linear -equations. [ABSTRACT FROM AUTHOR]
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- 2024
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10. A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments.
- Author
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Li, Yue-Wu and Qi, Feng
- Subjects
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GAUSSIAN function , *HYPERGEOMETRIC functions , *POWER series , *ARGUMENT , *INTEGRAL representations - Abstract
In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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11. Some Fourier transforms involving confluent hypergeometric functions.
- Author
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Berisha, Nimete Sh., Berisha, Faton M., and Fejzullahu, Bujar Xh.
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FOURIER transforms , *GAMMA functions , *INTEGRAL transforms , *HYPERGEOMETRIC functions , *MELLIN transform - Abstract
In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]
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- 2024
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12. New results on the associated Meixner, Charlier, and Krawtchouk polynomials.
- Author
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Ahbli, Khalid
- Subjects
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GENERATING functions , *POLYNOMIALS , *ORTHOGONAL polynomials , *HYPERGEOMETRIC functions , *HERMITE polynomials - Abstract
We give new explicit formulas as well as new generating functions for the associated Meixner, Charlier, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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13. Series representations for generalized harmonic functions in the case of three parameters.
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Klintborg, Markus
- Subjects
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STAR-like functions , *HARMONIC functions , *POWER series , *HYPERGEOMETRIC functions - Abstract
We present a canonical series expansion for generalized harmonic functions in the open unit disc in the complex plane that generalizes that recently obtained for the class of $ (p,q) $ (p , q) -harmonic functions. [ABSTRACT FROM AUTHOR]
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- 2024
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14. Fractional Calculus and Hypergeometric Functions in Complex Analysis.
- Author
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Oros, Gheorghe and Oros, Georgia Irina
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FRACTIONAL calculus , *HYPERGEOMETRIC functions , *ANALYTIC functions , *GEOMETRIC function theory , *HANKEL functions , *MEROMORPHIC functions , *SPECIAL functions - Abstract
This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]
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- 2024
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15. New Developments in Geometric Function Theory II.
- Author
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Oros, Georgia Irina
- Subjects
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GEOMETRIC function theory , *UNIVALENT functions , *ANALYTIC functions , *MEROMORPHIC functions , *SYMMETRIC functions , *HYPERGEOMETRIC functions , *INVERSE functions - Abstract
This document is a summary of a special issue of the journal Axioms titled "New Developments in Geometric Function Theory II." The special issue contains 14 research papers that explore various topics related to complex-valued functions in the field of Geometric Function Theory. The papers cover subjects such as coefficient estimates, subordination theories, hypergeometric functions, and differential operators. Each paper presents new findings and results that contribute to the development of Geometric Function Theory. The special issue is recommended for researchers and scholars interested in this field of study. The document also acknowledges the authors, reviewers, and editors involved in the creation of the special issue. [Extracted from the article]
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- 2024
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16. Proof of Two Supercongruences of Truncated Hypergeometric Series 4F3.
- Author
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Mao, Guo Shuai
- Subjects
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EULER number , *HYPERGEOMETRIC functions - Abstract
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf–Zeilberger method. One of them is, for any prime p > 3, 4 F 3 [ 7 6 1 2 1 2 1 2 1 6 1 1 | - 1 8 ] p - 1 2 = p ( - 2 p ) + p 3 4 ( 2 p ) E p - 3 (mod p 4) , where (· p) stands for the Legendre symbol, and En is the n-th Euler number. [ABSTRACT FROM AUTHOR]
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- 2024
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17. Solutions of the sl2${\mathfrak {sl}_2}$qKZ equations modulo an integer.
- Author
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Mukhin, Evgeny and Varchenko, Alexander
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INTEGERS , *DIFFERENCE equations , *EQUATIONS , *POLYNOMIALS , *HYPERGEOMETRIC functions , *DIOPHANTINE equations - Abstract
We study the qKZ difference equations with values in the n$n$th tensor power of the vector sl2${\mathfrak {sl}_2}$ representation V$V$, variables z1,⋯,zn$z_1,\dots,z_n$, and integer step κ$\kappa$. For any integer N$N$ relatively prime to the step κ$\kappa$, we construct a family of polynomials fr(z)$f_r(z)$ in variables z1,⋯,zn$z_1,\dots,z_n$ with values in V⊗n$V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of V⊗n$V^{\otimes n}$ are polynomials with integer coefficients. We show that fr(z)$f_r(z)$ satisfy the qKZ equations modulo N$N$. Polynomials fr(z)$f_r(z)$ are modulo N$N$ analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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18. Bilateral generating relations associated with two variable generalized hypergeometric polynomials.
- Author
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Bhagavan, V. S., Kameswari, P. L. Rama, and Srinivasulu, Tadikonda
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APPROXIMATION theory , *SPECIAL functions , *GENERATING functions , *RESEARCH personnel , *HYPERGEOMETRIC functions - Abstract
In this paper, the author first prove the theorem on bilateral generating relations for a certain two-variable generalised hypergeometric polynomials by the group theoretic technique introduced by Weisner. It is then shown how the main theorem can be applied to derive a large variety of bilateral generating functions for various special functions as well as for their various generalizations. Some results given by other researchers are thus observed to follow easily as special cases of the theorem proved in this paper. It is worth noting that special functions play role in the design of filters and approximation theory in communication engineering. [ABSTRACT FROM AUTHOR]
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- 2024
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19. A comment on the solutions of the generalized Faddeev–Volkov model.
- Author
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Dede, Mehmet
- Subjects
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YANG-Baxter equation , *HYPERBOLIC functions , *HYPERGEOMETRIC functions - Abstract
We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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20. On the modulo p zeros of modular forms congruent to theta series.
- Author
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Ringeling, Berend
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JACOBI forms , *EISENSTEIN series , *HYPERGEOMETRIC functions , *MODULAR groups , *ALGEBRAIC numbers , *MODULAR forms , *QUADRATIC fields - Abstract
For a prime p larger than 7, the Eisenstein series of weight p − 1 has some remarkable congruence properties modulo p. Those imply, for example, that the j -invariants of its zeros (which are known to be real algebraic numbers in the interval [ 0 , 1728 ]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight k ≥ 4 for the full modular group as the modular forms for which the first dim (M k) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j -invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. SOME NEW RESULTS ON FRACTIONAL INTEGRALS INVOLVING SRIVASTAVA POLYNOMIALS, (p,q)-EXTENDED HYPERGEOMETRIC FUNCTION AND M-SERIES.
- Author
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Sharma, Komal Prasad, Bhargava, Alok, and Agrawal, Garima
- Subjects
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FRACTIONAL integrals , *INTEGRAL operators , *GAUSSIAN function , *MATHEMATICAL domains , *HYPERGEOMETRIC functions , *POLYNOMIALS , *FRACTIONAL calculus - Abstract
Numerous prior publications on fractional calculus provide fascinating explanations of the theory and applications of fractional calculus operators throughout various mathematical analytic domains. In this paper, we introduce new fractional integral formulas using the Saigo-Maeda fractional integral operators and Appell’s function F3 along with the Srivastava polynomials, the (p,q)-extended Gauss hypergeometric function, and the M-Series. A few fascinating unusual cases of our main conclusions are also considered. This approach can be applied to explore a broad class of previously dispersed discoveries in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
22. A Note On Nielsen-Type Integrals, Logarithmic Integrals And Higher Harmonic Sums.
- Author
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Gupta, Bhawna, Qureshi, M. I., and Baboo, M. S.
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DEFINITE integrals , *HYPERGEOMETRIC functions , *INTEGRALS , *GENERALIZED integrals , *GAMMA functions , *ANALYTICAL solutions , *MELLIN transform - Abstract
Due to the great success of hypergeometric functions, we provide the analytical solutions of certain definite logarithmic integrals and Nielsen-type integrals in terms of multi-variable Kampé de Fériet functions with suitable convergence conditions and higher harmonic sums by using series rearrangement technique and incomplete Gamma function. Further we also obtain the solution of other related logarithmic integrals in terms of generalized hypergeometric functions and Kummer’s confluent hypergeometric functions by using series rearrangement technique. The results presented in the paper and comparable outcomes are hoped to be supplied by the use of computer-aid programs, for example, Mathematica. [ABSTRACT FROM AUTHOR]
- Published
- 2024
23. The distribution of the sample correlation coefficient under variance-truncated normality.
- Author
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Ogasawara, Haruhiko
- Subjects
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STATISTICAL correlation , *HYPERGEOMETRIC functions , *PROBABILITY density function , *BIVARIATE analysis - Abstract
The non-null distribution of the sample correlation coefficient under bivariate normality is derived when each of the associated two sample variances is subject to stripe truncation including usual single and double truncation as special cases. The probability density function is obtained using series expressions as in the untruncated case with new definitions of weighted hypergeometric functions. Formulas of the moments of arbitrary orders are given using the weighted hypergeometric functions. It is shown that the null joint distribution of the sample correlation coefficients under multivariate untruncated normality holds also in the variance-truncated cases. Some numerical illustrations are shown. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis.
- Author
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Cação, Isabel, Falcão, M. Irene, Malonek, Helmuth R., and Tomaz, Graça
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STURM-Liouville equation , *DIFFERENTIAL calculus , *SPHERICAL harmonics , *DIFFERENTIAL equations , *GENERATING functions , *HARMONIC analysis (Mathematics) , *HYPERGEOMETRIC functions - Abstract
The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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25. Deriving the Partial Fraction Series for the π cot(πx) Function by Applying Hypergeometric Functions Theory.
- Author
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López Carreño, Juan Carlos, Mendoza Suárez, Rosalba, and Mendoza Suárez, Jairo Alonso
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COTANGENT function , *HYPERGEOMETRIC functions , *HYPERGEOMETRIC series - Abstract
In this article, we present a new proof of the partial fraction expansion of the cotangent function using the theory of hypergeometric functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
26. ASYMPTOTIC ANALYSIS FOR CONFLUENT HYPERGEOMETRIC FUNCTION IN TWO VARIABLES GIVEN BY DOUBLE INTEGRAL.
- Author
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Yoshishige Haraoka
- Subjects
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HYPERGEOMETRIC functions , *INTEGRALS , *ASYMPTOTIC expansions - Abstract
We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along x = ∞ and y = ∞. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around (∞,∞). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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27. Investigation for the k-analogue of τ-Gauss hypergeometric matrix functions and associated fractional calculus.
- Author
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Abd-Elmageed, Hala, Hidan, Muajebah, and Abdalla, Mohamed
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MATRIX functions , *HYPERGEOMETRIC functions , *FRACTIONAL calculus , *INTEGRAL representations , *CALCULUS - Abstract
In this manuscript, we present a new definition of $ (k,\tau) $ (k , τ) -Gauss hypergeometric matrix function and study its analytical properties, like derivative formulas and integral representations. Furthermore, as an application we establish $ {\rm k} $ k -fractional calculus operators for the novel matrix function. We also give some new and known results as special cases of our proposed generalization of the Wright hypergeometric matrix function. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
28. Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix.
- Author
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Shimizu, Koki and Hashiguchi, Hiroki
- Subjects
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CHI-square distribution , *WISHART matrices , *EIGENVALUES , *DEGREES of freedom , *MATRIX functions - Abstract
This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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29. Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects.
- Author
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Aderyani, Safoura Rezaei, Saadati, Reza, O'Regan, Donal, and Alshammari, Fehaid Salem
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GAUSSIAN function , *HYPERGEOMETRIC functions , *GRONWALL inequalities , *FUZZY sets - Abstract
We apply known special functions from the literature (and these include the Fox H – function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the G – function, the Fox–Wright function and the Meijer G – function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
30. Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions.
- Author
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Bhowmik, Anwita and Barman, Rupam
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HYPERGEOMETRIC functions , *SUBGRAPHS , *FINITE fields - Abstract
For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2 t . Let g be a primitive element of the finite field F q . The Peisert graph P ∗ (q) is defined as the graph with vertex set F q where ab is an edge if and only if a - b ∈ ⟨ g 4 ⟩ ∪ g ⟨ g 4 ⟩ . We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in P ∗ (q) . We also give a new proof for the number of complete subgraphs of order three contained in P ∗ (q) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
31. New extensions of Hermite–Hadamard inequalities via generalized proportional fractional integral.
- Author
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Mumcu, İlker, Set, Erhan, Akdemir, Ahmet Ocak, and Jarad, Fahd
- Subjects
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FRACTIONAL integrals , *INTEGRAL inequalities , *CONVEX functions , *INTEGRAL operators , *HYPERGEOMETRIC functions - Abstract
The main aim this work is to give Hermite–Hadamard inequalities for convex functions via generalized proportional fractional integrals. We also obtained extensions of Hermite–Hadamard type inequalities for generalized proportional fractional integrals. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
- View/download PDF
32. New formulas of the high‐order derivatives of fifth‐kind Chebyshev polynomials: Spectral solution of the convection–diffusion equation.
- Author
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Abd‐Elhameed, Waleed M. and Youssri, Youssri H.
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TRANSPORT equation , *CHEBYSHEV polynomials , *HYPERGEOMETRIC functions , *POLYNOMIALS - Abstract
This paper is dedicated to deriving novel formulae for the high‐order derivatives of Chebyshev polynomials of the fifth‐kind. The high‐order derivatives of these polynomials are expressed in terms of their original polynomials. The derived formulae contain certain terminating 4F3(1) hypergeometric functions. We show that the resulting hypergeometric functions can be reduced in the case of the first derivative. As an important application—and based on the derived formulas—a spectral tau algorithm is implemented and analyzed for numerically solving the convection–diffusion equation. The convergence and error analysis of the suggested double expansion is investigated assuming that the solution of the problem is separable. Some illustrative examples are presented to check the applicability and accuracy of our proposed algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Summation formulas generated by Hilbert space eigenproblem.
- Author
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Mali, Petar, Gombar, Sonja, Radošević, Slobodan, Rutonjski, Milica, Pantić, Milan, and Pavkov-Hrvojević, Milica
- Subjects
- *
INFINITE series (Mathematics) , *HYPERGEOMETRIC functions , *QUANTUM mechanics , *POTENTIAL well , *HILBERT space - Abstract
We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. 3 F 4 Hypergeometric Functions as a Sum of a Product of 2 F 3 Functions.
- Author
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Straton, Jack C.
- Subjects
- *
HYPERGEOMETRIC functions , *WHITTAKER functions , *HYPERGEOMETRIC series - Abstract
This paper shows that certain 3 F 4 hypergeometric functions can be expanded in sums of pair products of 2 F 3 functions, which reduce in special cases to 2 F 3 functions expanded in sums of pair products of 1 F 2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2 F 1 functions, and 3 F 2 functions into the realm of p F q functions where p < q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. Four Families of Summation Formulas for 4 F 3 (1) with Application.
- Author
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Kumar, Belakavadi Radhakrishna Srivatsa, Rathie, Arjun K., and Choi, Junesang
- Subjects
- *
PARTITION functions , *HYPERGEOMETRIC series , *HYPERGEOMETRIC functions , *GAMMA functions , *BETA functions , *INTEGERS - Abstract
A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Single-Shot Factorization Approach to Bound States in Quantum Mechanics.
- Author
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Mazhar, Anna, Canfield, Jeremy, Mathews Jr., Wesley N., and Freericks, James K.
- Subjects
- *
BOUND states , *EIGENVALUES , *HYPERGEOMETRIC functions , *FACTORIZATION , *SCHRODINGER equation , *DIFFERENTIAL equations , *EIGENVECTORS - Abstract
Using a flexible form for ladder operators that incorporates confluent hypergeometric functions, we show how one can determine all of the discrete energy eigenvalues and eigenvectors of the time-independent Schrödinger equation via a single factorization step and the satisfaction of boundary (or normalizability) conditions. This approach determines the bound states of all exactly solvable problems whose wavefunctions can be expressed in terms of confluent hypergeometric functions. It is an alternative that shares aspects of the conventional differential equation approach and Schrödinger's factorization method, but is different from both. We also explain how this approach relates to Natanzon's treatment of the same problem and illustrate how to numerically determine nontrivial potentials that can be solved this way. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. On Abel's Problem and Gauss Congruences.
- Author
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Delaygue, É and Rivoal, T
- Subjects
- *
ALGEBRAIC functions , *PRIME numbers , *HYPERGEOMETRIC series , *DIFFERENTIAL equations , *HYPERGEOMETRIC functions , *ARITHMETIC , *GEOMETRIC congruences - Abstract
A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|. Risch designed an algorithm that, given |$\eta $| , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$| , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Log-concavity and log-convexity of series containing multiple Pochhammer symbols.
- Author
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Karp, Dmitrii and Zhang, Yi
- Subjects
- *
POWER series , *GENERIC products , *SPECIAL functions , *SIGNS & symbols , *HYPERGEOMETRIC functions - Abstract
In this paper, we study power series with coefficients equal to a product of a generic sequence and an explicitly given function of a positive parameter expressible in terms of the Pochhammer symbols. Four types of such series are treated. We show that logarithmic concavity (convexity) of the generic sequence leads to logarithmic concavity (convexity) of the sum of the series with respect to the argument of the explicitly given function. The logarithmic concavity (convexity) is derived from a stronger property, namely, positivity (negativity) of the power series coefficients of the so-called generalized Turánian. Applications to special functions such as the generalized hypergeometric function and the Fox-Wright function are also discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Plea for Diagonals and Telescopers of Rational Functions.
- Author
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Hassani, Saoud, Maillard, Jean-Marie, and Zenine, Nadjah
- Subjects
- *
ALGEBRAIC functions , *HYPERGEOMETRIC functions , *ISING model , *ELLIPTIC functions , *TRANSCENDENTAL functions - Abstract
This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P / Q k and 1 / Q , are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple F 1 2 hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. GHOSTS AND CONGRUENCES FOR $\boldsymbol {p}^{\boldsymbol {s}}$ -APPROXIMATIONS OF HYPERGEOMETRIC PERIODS.
- Author
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VARCHENKO, ALEXANDER and ZUDILIN, WADIM
- Subjects
- *
HYPERGEOMETRIC functions , *ANALYTIC functions , *LAURENT series , *ARITHMETIC , *POLYNOMIALS - Abstract
We prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p -adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p -adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Generalised unitary group integrals of Ingham-Siegel and Fisher-Hartwig type.
- Author
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Akemann, Gernot, Aygün, Noah, and Würfel, Tim R.
- Subjects
- *
UNITARY groups , *HAAR integral , *INTEGRALS , *HYPERGEOMETRIC functions , *EIGENVALUES , *MATHEMATICS , *DETERMINANTS (Mathematics) - Abstract
We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group U(N) with respect to Haar measure, for finite N and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, such integrals can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauß' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg [J. Phys. A: Math. Theor. 51(34), 345202 (2018)]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Finite Representations of the Wright Function.
- Author
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Prodanov, Dimiter
- Subjects
- *
SPECIAL functions , *ERROR functions , *HEAT equation , *ALGEBRA , *AIRY functions , *HYPERGEOMETRIC functions - Abstract
The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1 F 2 Hypergeometric Functions That Arise from Them.
- Author
-
Straton, Jack C.
- Subjects
- *
HYPERGEOMETRIC functions , *INFINITE series (Mathematics) , *ANGLES , *POLYNOMIAL approximation , *BESSEL functions - Abstract
The Bessel function of the first kind J N k x is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I N k x . The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for J N k x useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving 1 F 2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Bi-Concave Functions Connected with the Combination of the Binomial Series and the Confluent Hypergeometric Function.
- Author
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Srivastava, Hari M., El-Deeb, Sheza M., Breaz, Daniel, Cotîrlă, Luminita-Ioana, and Sălăgean, Grigore Stefan
- Subjects
- *
HYPERGEOMETRIC functions , *HYPERGEOMETRIC series , *UNIVALENT functions , *ANALYTIC functions , *GAUSSIAN function - Abstract
In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. On the Analytic Extension of Lauricella–Saran's Hypergeometric Function F K to Symmetric Domains.
- Author
-
Dmytryshyn, Roman and Goran, Vitaliy
- Subjects
- *
SYMMETRIC domains , *CONTINUED fractions , *HYPERGEOMETRIC functions , *FUNCTIONS of several complex variables , *ANALYTIC functions , *SPECIAL functions - Abstract
In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran's hypergeometric function F K with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Results concerning multi-index Wright generalized Bessel function.
- Author
-
Khan, Nabiullah and Iqbal Khan, Mohammad
- Subjects
- *
WHITTAKER functions , *BESSEL functions , *MELLIN transform , *LAGUERRE polynomials , *BETA functions , *HYPERGEOMETRIC functions , *INTEGRAL transforms - Abstract
In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Some finite integrals involving Mittag-Leffler confluent hypergeometric function.
- Author
-
Pal, Ankit
- Subjects
- *
SPECIAL functions , *INTEGRALS , *HYPERGEOMETRIC functions - Abstract
In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. On Algebraic Properties of Integrals of Products of Some Hypergeometric Functions.
- Author
-
Gorelov, V. A.
- Subjects
- *
DIFFERENTIAL equations , *INTEGRALS , *GENERALIZED integrals , *HYPERGEOMETRIC functions - Abstract
Indefinite integrals of products of generalized hypergeometric functions satisfying first- order differential equations are considered. Necessary and sufficient conditions for the algebraic independence of the set of these integrals and of their values at algebraic points are studied. All algebraic identities arising in this case are found in closed form. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Cycles on Jacobians of Fermat curves and hypergeometric functions.
- Author
-
Sarkar, Subham
- Subjects
- *
JACOBIAN matrices , *HYPERGEOMETRIC functions , *HYPERGEOMETRIC series - Abstract
In this paper we construct certain higher Chow cycles in K 1 of the Jacobian of Fermat curves, generalising a construction of Collino. Then we express the regulator of these elements in terms of the special values of hypergeometric functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Hypergeometric identities related to Ruijsenaars systems.
- Author
-
Belousov, N., Derkachov, S., Kharchev, S., and Khoroshkin, S.
- Subjects
- *
OPERATOR theory , *KERNEL functions , *HYPERGEOMETRIC functions - Abstract
We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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