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Abstract We prove the sharp weighted trace Hardy inequality on the half space in the Finsler context. The obtained inequality extends the results obtained in [23,29] in the Euclidean context to the Finsler context. It also generalizes the recent result in [5] to the weight context. [ABSTRACT FROM AUTHOR]

Let E ( r ) be the complete elliptic integrals of the second kind and J p ( a , b ) be the one-parameter mean of two distinct positive numbers a and b defined by J p ( a , b ) = p p + 1 a p + 1 − b p + 1 a p − b p for p ≠ − 1 , 0 . In this study using the recurrence method with the sign rule of a special power series, we prove that the double inequality 7 π 22 J 2 / ( π − 2 ) ( 1 , r ′ ) < 2 π E ( r ) < J 2 / ( π − 2 ) ( 1 , r ′ ) holds for r ′ = 1 − r 2 ∈ ( 0 , 1 ) , where the upper bound is sharp, while the lower bound is related to the Archimedes' approximation 22/7 for π . Using the same method, we also easily show that both of the functions r 2 ↦ ( 2 / π ) E ( r ) − J 7 / 4 ( 1 , r ′ ) r 12 and r 2 ↦ 1 r 10 ( 1 − 1 − ( 2 / π ) E ( r ) 1 − J 7 / 4 ( 1 , r ′ ) ) are absolutely monotonic on ( 0 , 1 ) , which greatly improves the known results. Moreover, it should be emphasized that our method may be used to explore special functions related to hypergeometric functions. [ABSTRACT FROM AUTHOR]

Zudilin proved some Ramanunjan-type supercongruences by the WZ method. Long proved a Ramanujan-type supercongruence conjecture due to Van Hamme by applying some hypergeometric evaluation identities. In this paper, we propose a conjecture on a complete q -analogue of this supercongruence of Van Hamme and prove it in a weaker form via the q -WZ method. Additionally, we give a q -analogue of a related congruence involving cubes of binomial coefficients due to Sun in the same way. We also present a conjecture on a q -analogue of the corresponding infinite series for 1 / π due to Ramanujan. [ABSTRACT FROM AUTHOR]

We have used recent integral representations of the derivatives of the Bessel functions with respect to the order to obtain closed-form expressions in terms of generalized hypergeometric functions and Meijer- G functions. Also, we have carried out similar calculations for the derivatives of the modified Bessel functions with respect to the order, obtaining closed-form expressions as well. For this purpose, we have obtained integral representations of the derivatives of the modified Bessel functions with respect to the order. As by-products, we have calculated two non-tabulated integrals. [ABSTRACT FROM AUTHOR]

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann Ξ-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Ξ-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem. [ABSTRACT FROM AUTHOR]

Any three basic hypergeometric series ϕ 1 2 whose respective parameters, a , b , and c , differ by integer powers of the base q satisfy a linear relation with coefficients that are rational functions of a , b , c , q , and the variable x . These relations are called three-term relations for the basic hypergeometric series ϕ 1 2 . This paper gives explicit expressions for the coefficients of these three-term relations. [ABSTRACT FROM AUTHOR]

Abstract For the hypergeometric function of unit argument F 2 3 (1) we prove the existence and uniqueness of three-term contiguous relations with arbitrary integer shifts. We show that not only the original F 2 3 (1) function but also other five functions related to it satisfy one and the same three-term relation. This fact together with the uniqueness mentioned above provides three-term relations with a group symmetry of order 72. [ABSTRACT FROM AUTHOR]

In this work, we derive relations between generating functions of double stuffle relations and double shuffle relations to express the alternating double Euler sums ζ ( r ‾ , s ) , ζ ( r , s ‾ ) and ζ ( r ‾ , s ‾ ) with r + s odd in terms of zeta values. We also give a direct proof of a hypergeometric identity which is a limiting case of a basic hypergeometric identity of Andrews. Finally, we gave another proof for the formula of Zagier on the multiple zeta values ζ ( 2 , … , 2 , 3 , 2 , … , 2 ) . [ABSTRACT FROM AUTHOR]

The paper studies logarithmic convexity and concavity of power series with coefficients involving q -gamma functions or q -shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such series are basic hypergeometric functions. We consider four types of series. For each type we establish conditions sufficient for the power series coefficients of the generalized Turánian formed by these series to have constant sign. Finally, we furnish seven examples of basic hypergeometric functions satisfying our general theorems. This investigation extends our previous results on power series with coefficient involving the ordinary gamma functions and the shifted factorials. [ABSTRACT FROM AUTHOR]

A new generalization of the modified Bessel function of the second kind K z ( x ) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos ⁡ ( π z ) M 2 z ( 4 x ) − sin ⁡ ( π z ) J 2 z ( 4 x ) and which subsumes the self-reciprocal pair involving K z ( x ) . Its application towards finding modular-type transformations of the form F ( z , w , α ) = F ( z , i w , β ) , where α β = 1 , is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on S L 2 ( Z ) . This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

We first construct a new discrete probability space ( Ω , F , P ) and then define a random variable with probability distribution W ( x , y ; q ) in this space. As applications, we give a probabilistic form of Al-Salam and Verma q -integral, a new q -identity of ϕ 2 3 and the two terms ϕ 2 3 transformation formula of Sears. [ABSTRACT FROM AUTHOR]

A simple uniform bound for the ratio of two modified spherical Bessel functions is derived. The arguments of the two functions are complex but their ratio is real. [ABSTRACT FROM AUTHOR]

We show how to get Krall polynomials from Krall discrete polynomials using a procedure of passing to the limit in some of the parameters of the family. We also show that this procedure has to be different to the standard one used in the Askey scheme to go from the classical discrete polynomials to the classical polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that ∑ k = 0 p − 1 ( p − 1 k ) ( 2 k k ) 2 P k 8 k ≡ 0 ( mod p 2 ) for any prime p ≡ 7 ( mod 8 ) , where P k is the k -th Pell number. Further, we also propose three new congruences of the same type. [ABSTRACT FROM AUTHOR]

In this paper we consider a master integral in four arbitrary parameters. The integrand involves the logarithmic function and the Gauss hypergeometric function, which in certain special cases the integral reduces to identities involving zeta functions. A relationship will also be created between the integral and Euler sums of arbitrary order and arbitrary argument. Many interesting new specific examples will be highlighted. [ABSTRACT FROM AUTHOR]

In this paper, we define the function R q p ( α , β ; z ) and obtain its properties including analytic properties (type and order), contiguous relations, differential property and simple integrals. We also establish its relation with other well known special functions. [ABSTRACT FROM AUTHOR]

Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In this paper we provide an alternative way of obtaining such results. Our method is very simple and it is based on the non-local derangement identity. [ABSTRACT FROM AUTHOR]

New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed persymmetric Jacobi matrix. The orthogonality relation and an explicit expression in terms of hypergeometric functions are also given. Special cases and connections with the para-Krawtchouk polynomials and the dual-Hahn polynomials are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the double inequality λ S 11 / 4 , 7 / 4 ( 1 , r ′ ) < E ( r ) < μ S 11 / 4 , 7 / 4 ( 1 , r ′ ) holds for all r ∈ ( 0 , 1 ) if and only if λ ≤ π / 2 = 1.570796 ⋯ and μ ≥ 11 / 7 = 1.571428 ⋯ , where r ′ = ( 1 − r 2 ) 1 / 2 , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ⁡ ( t ) d t is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 / ( p − q ) is the Stolarsky mean of a and b . [ABSTRACT FROM AUTHOR]

We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the two-dimensional case. [ABSTRACT FROM AUTHOR]

Within the Askey scheme of hypergeometric orthogonal polynomials, Racah polynomials stay on the top of the hierarchy and they generalize all of the discrete hypergeometric orthogonal polynomials. In this paper, we investigate asymptotic behaviors of Racah polynomials with varying parameters when the polynomial degree tends to infinity. Using the difference equation technique developed in our earlier papers, we obtain an asymptotic formula in the outer region via ratio asymptotics and then derive asymptotic formulas in the oscillatory region via a matching method. Our asymptotic formulas are explicitly given in terms of the polynomial degree, variable and parameters, using elementary functions such as logarithmic, exponential and rational functions. By taking limits, our results also yield asymptotic formulas for orthogonal polynomials in the lower hierarchy of the Askey scheme such as Hahn polynomials and Krawtchouk polynomials. [ABSTRACT FROM AUTHOR]

The generalized hypergeometric function satisfies many identities or “transforms” which can be used to establish their asymptotic behavior for large argument and even, in some cases, for large parameters. We will show that using just three transforms alone, valid for a large class of multivariate hypergeometric functions, we can use a similar “calculus” to compute asymptotic expansions even in higher dimensions. [ABSTRACT FROM AUTHOR]

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure, i.e., 1 x − c x α e − x d x + N δ ( x − c ) , for x ∈ [ 0 , ∞ ) , α > − 1 , a free parameter N ∈ R + and a shift c < 0 . We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as n tends to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the double inequality 1 + α r ′ 2 < K a ( r ) sin ⁡ ( π a ) log ⁡ ( e R ( a ) / 2 / r ′ ) < 1 + β r ′ 2 holds for all a ∈ ( 0 , 1 / 2 ] and r ∈ ( 0 , 1 ) if and only if α ≤ π / [ R ( a ) sin ⁡ ( π a ) ] − 1 and β ≥ a ( 1 − a ) , where r ′ = 1 − r 2 , K a ( r ) is the generalized elliptic integral of the first kind and R ( x ) is the Ramanujan constant function. Besides, as the key tool, the series expression for the Ramanujan constant function R ( x ) is given. [ABSTRACT FROM AUTHOR]

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As applications, equivalent forms, the reverses, some particular examples, two kinds of Hardy-type inequalities, and operator expressions are considered. [ABSTRACT FROM AUTHOR]

We use connection relations and series rearrangement to generalize generating functions for several higher continuous orthogonal polynomials in the Askey scheme, namely the Wilson, continuous dual Hahn, continuous Hahn, and Meixner–Pollaczek polynomials. We also determine corresponding definite integrals using the orthogonality relations for these polynomials. [ABSTRACT FROM AUTHOR]

We give a general expansion formula of functions in the Askey–Wilson polynomials and using Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q -Jacobi type polynomials. We also give a new generating function for Askey–Wilson polynomials and a new evaluation for specialized Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

This paper obtains several evaluations of multivariate hypergeometric functions for particular parameter values and at special algebraic points. They have a high interest not only on their own, but also in the light of the remarkable implications for both pure mathematics and mathematical physics. Following our research started in [30] and [31], we provide some contribution to such functions' computability inside and outside their disks of convergences. In the first part we provide some new results in the spirit of Theorem 3.1 of [31], obtaining formulae for the values of multivariate hypergeometric functions by generalizing a well known identity of Kummer [23], to the hypergeometric functions of two or more variable like those of Appell and Lauricella denoted FD(n). In the second part, using some reduction schemes of hyperelliptic integrals due to Goursat [16], Hermite [18,19] we evaluate Appell and Lauricella's FD(n) hypergeometric functions and their analytic continuations at some particular locations. Finally, by exploiting reductions of hyperelliptic integrals to elliptic due to Belokolos et al. [5], Eilbeck and Enol'skii [11], Enol'skii and Kostov [12] and by Maier [27], we obtain further links from multivariate hypergeometric functions, to complete elliptic integrals and to π. We thus reach a conceptual settlement of the piece of research started by us in [30] and [31]. [ABSTRACT FROM AUTHOR]

We define the generalized hypergeometric polynomial of degree N as follows: P N ( α 1 , . . . , α p ; β 1 , . . . , β q ; z ) = ∑ m = 0 N [ ( − N ) m ( α 1 ) m ⋅ ⋅ ⋅ ( α p ) m z N − m m ! ( β 1 ) m ⋅ ⋅ ⋅ ( β q ) m ] = z N F q p + 1 ( − N , α 1 , . . . , α p ; β 1 , . . . , β q ; 1 / z ) . Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p + q parameters α j and β k are arbitrary (“generic”, possibly complex) numbers, ( α ) m is the Pochhammer symbol and F q p + 1 ( α 0 , α 1 , . . . , α p ; β 1 , . . . , β q ; z ) is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζ n of this polynomial. We moreover manufacture an N × N matrix L ̲ in terms of the 1 + p + q parameters N , α j , β k characterizing this polynomial, and of its N zeros ζ n , and we show that it features the N eigenvalues λ m = m ∏ k = 1 q ( − β k + 1 − m ) , m = 1 , . . . , N . These N eigenvalues depend only on the q parameters β k , implying that the N × N matrix L ̲ is isospectral for variations of the p parameters α j ; and they clearly are integer (or rational ) numbers if the q parameters β k are themselves integer (or rational ) numbers: a nontrivial Diophantine property. [ABSTRACT FROM AUTHOR]

Abstract: By two Slater's hypergeometric series identities, we establish two general summation formulas with six free parameters. Specializing certain parameters in these formulas, we obtain a list of Ramanujan-type series formulas for π, and . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples. [Copyright &y& Elsevier]

Abstract: We provide the full asymptotic expansion of the harmonic Berezin transform on the unit ball in purely by means of transformations of hypergeometric functions and functionʼs “hypergeometrization”. [Copyright &y& Elsevier]

Abstract: We construct classes of Bailey pairs where the exponent of q in is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums. [Copyright &y& Elsevier]

Abstract: The Goursat formula for the hypergeometric function extends the Euler–Gauss relation to the case of logarithmic singularities. We study the monodromic functional equation associated with a perturbation of the Bessel differential equation by means of a variant of the Laplace–Borel technique: we introduce and study a related monodromic equation in the dual complex plane. This construction is a crucial element in our proof of a duality theorem that leads to an extension of the Euler–Gauss–Goursat formula for hypergeometric functions to a substantially larger class of functions. [Copyright &y& Elsevier]

Abstract: We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals. [Copyright &y& Elsevier]

Abstract: We concretely construct the solution of the singular Cauchy problem for a Fuchsian partial differential operator of weight one by means of hypergeometric representations and study what kind of singularities appear on the characteristic surfaces determined by the principal part of the operator. In particular we investigate how the singularities are characterized not only by the principal part but also by the subprincipal part restricted at the origin and the ramification index of Cauchy data. [Copyright &y& Elsevier]

Abstract: We present a general theory for studying the difference analogs of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference–recurrence relations for such functions. Finally, applications to -classical polynomials are given. [Copyright &y& Elsevier]

Abstract: Classical integral representation of the Mellin type kernel in terms of the Laplace integral gives an idea to construct a class of non-convolution (index) transforms with the kernel where admits a power series expansion, which has an infinite radius of convergence and the integral converges absolutely in a half-plane of the complex plane . Particular examples give the Kontorovich–Lebedev-like transformation and new transformations with hypergeometric functions as kernels. Mapping properties and inversion formulas are obtained. Finally we prove a new inversion theorem for the modified Kontorovich–Lebedev transform. [Copyright &y& Elsevier]

Abstract: The Sears transformations for non-terminating -series are applied to derive several general series transformations for another two double -Clausen series of type and . Further new transformations and reduction formulae are established for the basic Clausen hypergeometric series. In particular, several bibasic reduction formulae are obtained. [Copyright &y& Elsevier]

Abstract: By using a generalization of Sturm–Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal polynomials, is introduced. The standard properties of these polynomials, such as a second order difference equation, an explicit form for the polynomials, a three term recurrence relation and an orthogonality relation are presented. It is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class. Moreover, moments corresponding to these weight functions can be explicitly computed. Finally, a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail. [Copyright &y& Elsevier]

Abstract: In terms of the hypergeometric method, we establish ten general -formulas with free parameters which include several known results as special cases. [Copyright &y& Elsevier]

Abstract: In this paper, we apply generalized Stieltjes transform representation to study the generalized hypergeometric function. Among the results thus proved are new integral representations, inequalities, properties of the Padé table and the properties of the generalized hypergeometric function as a conformal map. [Copyright &y& Elsevier]

Abstract: Using some properties of the general rising shifted factorial and the gamma function we derive a variant form of Dougallʼs summation for the classical hypergeometric functions. This variant form allows us to derive easily many Ramanujan type series for and Ramanujan type series for some other constants. [Copyright &y& Elsevier]

Abstract: In this paper we derive a family of asymptotic expansions for the nth Landau constant . The proof is based on a new general integral representation of the constants in terms of a Hypergeometric Function. As a consequence we prove two conjectures related to earlier asymptotic developments of the ʼs. [Copyright &y& Elsevier]

Abstract: In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed. [Copyright &y& Elsevier]

Abstract: In this paper we consider a function , which can be written as a linear combination of two Saalschützian hypergeometric series or as a very-well-poised hypergeometric series. We explore two-term and three-term relations satisfied by the L function and put them in the framework of group theory. We prove a fundamental two-term relation satisfied by the L function and show that this relation implies that the Coxeter group , which has 1920 elements, is an invariance group for . The invariance relations for are given two classifications based on two double coset decompositions of the invariance group. The fundamental two-term relation is shown to generalize classical results about hypergeometric series. We derive Thomaeʼs identity for series, Baileyʼs identity for terminating Saalschützian series, and Barnesʼ second lemma as consequences. We further explore three-term relations satisfied by . The group that governs the three-term relations is shown to be isomorphic to the Coxeter group , which has 23 040 elements. Based on the right cosets of in , we demonstrate the existence of 220 three-term relations satisfied by the L function that fall into two families according to the notion of L-coherence. The complexity of the coefficients in front of the L functions in the three-term relations is studied and is shown to also depend on L-coherence. [Copyright &y& Elsevier]

Abstract: In this paper, we study the relationship between the generalized Hersch–Pfluger distortion function and complete elliptic integrals of the second kind , and establish several inequalities for and . [Copyright &y& Elsevier]

Abstract: We undertake a thorough investigation of the moments of Ramanujanʼs alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities. [Copyright &y& Elsevier]