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In the paper, the author presents bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions.

International audience; In the paper, the author investigates some properties, including analyticity, limits, monotonicity, complete monotonicity, and inequalities, of several functions involving the tri-, tetra-, and penta-gamma functions and originating from computation of the sectional curvature of the beta manifold.

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine ( &alpha, m ) &isin, R + &times, N such that the function x &alpha, &psi, ( m ) ( x ) | is completely monotonic on ( 0 , &infin, ) , where &psi, ( x ) denotes the logarithmic derivative of Euler&rsquo, s gamma function.

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol&ndash, Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis’s formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function.

In the paper, we establish necessary and sufficient conditions for two families of functions involving divided differences of the di- and tri-gamma functions to be completely monotonic. Consequently, we derive necessary and sufficient conditions for two families of functions involving the ratio of two gamma functions to be logarithmically completely monotonic. Furthermore, we deduce some inequalities for bounding the ratio of two gamma functions and divided differences of polygamma functions.

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl. 2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions.

If f is a non-constant meromorphic function such that a power of f shares certain set counting multiplicities in the weak sense with the derivative of the power, then f becomes an exponential function.

In the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

In the paper we discuss the value distribution of the product of the derivative of a transcendental meromorphic function and a power of the function.

Some complete monotonicity results for q -polygamma functions are proved. Our results extend positivity of some functions containing q -polygamma functions to complete monotonicity property. Also, we give two new inequalities for q -trigamma function.

In the paper, the authors prove that the functions |?(i)(ex)| for i ? N are subadditive on (ln?i,?) and superadditive on (-?, ln ?i), where ?i ? (0,1) is the unique root of equation 2|?(i)(?)|= ?(i)(?2)|.

We give a purely algebraic proof of a formula for Taylor coefficients of the reciprocal gamma function. The formula expresses each coefficient in terms of multiple zeta values. Our proof uses Hoffman’s harmonic algebra of multiple zeta values.

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemannian zeta function, and the alternative Hurwitz zeta function. Keywords: Series identity, Catalan number, Catalan function, Riemannian zeta function, Alternative Hurwitz zeta function, Digamma function

In this paper, we rederive the identity Γ q (x)Γ q (1 −x) = π q sin q (π q x) . Then, we give q-analogue of Gauss’ multiplication formulaand study representation of q-oscillator algebra in terms of the q-factorialpolynomials. 1. IntroductionIn the last decades, the q-calculus served as a bridge between mathematicsand physics. The majority of researchers around the world who use q-calculusare physicists. This field has expanded explosively, due to the fact that thebasic hypergeometric series served several subjects of combinatorics, quantumtheory, number theory, statistical mechanics.From now on we will restrict our concern to the case that the deformationparameter qis real and 0

We present three formulas for summation in closed form (or certain transformation) of series constructed with binomial expressions. Examples and application are provided, involving the Hurwitz zeta function, the dilogarithm, and the digamma function.

This paper deals with the [Formula: see text]-analogues of generalized zeta matrix function, digamma matrix function and polygamma matrix function. We also discuss their regions of convergence, integral representations and matrix relations obeyed by them. We also give a few identities involving digamma matrix function and [Formula: see text]-hypergeometric matrix series.

We present new asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions. Based on these expansions, we prove new complete mono- tonicity properties of some functions involving the gamma and polygamma functions. As con- sequences of them we establish new upper and lower bounds for the gamma function in terms of the polygamma functions.

We extend some known inequalities for the digamma function to the q-digamma function. Furthermore, we estimate the unique positive real root of the q-digamma function @j"q.

Necessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

In this paper, we present two inequalities involving the product of the gamma function and the Hurwitz zeta function. Mathematics Subject Classification: 26D15

In this paper, we derive a class of approximations of the $q$-digamma function $\psi _q(x)$. The infinite family \[ I_a(x;q)=\log [x+a]_q+\frac {q^x\log q}{1-q^x}-\bigg (\frac 12-a\bigg )H(q-1)\log q, \] $a\in [0,1]$; $q>0$, can be used as approximating functions for $\psi _q(x)$, where $[x]_q=(1-q^x)/(1-q)$ and $H(\cdot )$ is the Heaviside step function. We show that, for all $a\in [0,1]$, $I_a$ is asymptotically equivalent to $\psi _q(x)$ for $q>0$ and is a good pointwise approximation.

In the paper, some lower bounds for polygamma functions are refined., 8 pages

In this paper we shall locate a class of fundamental identities for the gamma function and trigonometric functions in the chart of functional equations for the zeta- functions as a manifestation of the underlying modular relation. We use the beta-transform but not the inverse Heaviside integral. Instead we appeal to the reciprocal relation for the Euler digamma function which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the gamma function through the digamma function.

The Gamma function and its n th logarithmic derivatives (that is, the polygamma or the psi-functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi-functions and to evaluate a class of log-sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi-function expressions for some known parameterized multiple sums (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We prove a class of double inequalities for the gamma function which were conjectured by Batir [On some properties of the gamma function, Expo. Math. 26 (2008) 187–196].

Some inequalities for the gamma function are given. These results refine the classical Stirling approximation and its many recent improvements.

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.

The function [Γ(x+1)]1/x x ( 1 + 1 x )x is strictly logarithmically completely monotonic in (0,∞). The function ψ′′(x + 2) + 1+x 2 x2(1+x)2 is strictly completely monotonic in (0,∞).

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function.

The function [Γ(x + 1)]1/x(1 + 1/x)x/x is strictly logarithmically completely monotonic in (0, ∞). The function ψ″ (x + 2) + (1 + x2)/x2(1 + x)2 is strictly completely monotonic in (0, ∞).

By employing the convolution theorem of Laplace transforms, some asymptotic formulas and integral representations of the gamma, psi and polygamma functions, and other analytic techniques, this note provides an alternative proof of a monotonicity and convexity property by N. Elezovic, C. Giordano and J. Pecaric in [4] to establish the best bounds in Gautschi- Kershaw inequalities. Moreover, some (logarithmically) complete monotonicity results on functions related to Gautschi-Kershaw inequalities are remarked.

On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.

The logarithmic derivative of the $\Gamma$-function, namely the $\psi$-function, has numerous applications. We define analogous functions in a four dimensional space. We cut lattices and obtain Clifford-valued functions. These functions are holomorphic cliffordian and have similar properties as the $\psi$-function. These new functions show links between well-known constants: the Euler gamma constant and some generalisations, $\zeta^R(2)$, $\zeta^R(3)$. We get also the Riemann zeta function and the Epstein zeta functions.

We define a q-analogue of the basic multiple gamma function [Formula: see text] introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m + 1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.

In this paper, we establish an approximate functional equation for the Lerch zeta function, which is a generalization of the Riemann zeta function and the Hurwitz zeta function.

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or$f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.AMS 2000 Mathematics subject classification: Primary 30D35

The zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

In the paper, a discrete distribution of the Matsumoto zeta‐function is considered. It is proved that the probability measure \(P_N (A) = {\mu}_N (\phi (s + ikh) \in A),{\text{ }}A \in \mathcal{B}(H(D))\), converges weakly as \(N \to \infty \).

Let be the Hurwitz zeta function. Furthermore, let p > 1 and α ≠ 0 be real numbers and n ≥ 2 be an integer. We determine the best possible constants a(p, α, n), A(p, α, n), b(p, n) and B(p, n) such that the inequalities and hold for all positive real numbers x1,…,xn.

Some basic comparaison theorems between two elementary means — differential and integral f -mean are obtained. This theorems are applied to digamma function. Mathematics subject classification (1991): 26D10, 33B15.

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous nth_order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order n x n systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determinod. Connections with completely integrable systems are mentioned. In order to set the stage for our extension of Picard's theorem on the existence of elliptic solutions of the second kind of nth_order scalar ordinary differential equations with elliptic coefficients (with a common period lattice) to first-order n x n systems and an explicit description of the corresponding Floquet-type structure of fundamental systems of solutions, we briefly review Floquet theory for singly periodic n x n systems. Denote by M(n), n E N, the set of n x n matrices with entries in C, define GL(n) {A E M(n) I det(A) 7 0}, and consider the linear homogeneous system (1) @'I(z) Q(z)'J'(z), z E C, where @(z) E GL(n), Q(z) E M(n), with Q(z) a continuous periodic matrix of period Q E C\{0}, that is, (2) Q(z+Q) = Q(z), z E C. Concerning (continuously differentiable) fundamental matrices 4>(z) of solutions of (1), one has the following basic Floquet theorem (see, e.g., [5], Ch. 3, [20], Ch. 4, [30], Ch. 5). Theorem 1. Let Q(z) E M(n) with Q(z) a continuous periodic matrix of period Q E C\{0}. Then (1) admits a fundamental matrix 4?(z) E GL(n) of the type (3) 4(z) = P(z) exp (zK), z E C, where P(z) E GL(n), K E M(n), and P(z) is continuously differentiable and periodic of period Q, (4) P(z+Q) = P(z), z E C. Received by the editors September 23, 1996. 1991 Mathematics Subject Classification. Primary 33E05, 34C25; Secondary 58F07. The research was based upon work supported by the National Science Foundation under Grant No. DMS-9623121. (g)1998 American Mathematical Society

The following inequality relating to the ratio of the gamma functions α logx x− Γ(x) Γ ( x+ x ) , where α is a suitable constant, is established for every x > 0 . This inequality gives a contribution to the recent results, proved by several authors, involving the functions Γ(x) and Γ ( 1 x ) . It also gives an alternative proof of a conjecture formulated by D. Kershaw and recently proved by G.J.O. Jameson and T.P. Jameson [5]. Mathematics subject classification (2010): 33B15, 26D07.

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.