The multiple gamma function $Γ_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $Γ_n$ function and their applications to summation of series and infinite products.
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 exten…
Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this article we exploit the concepts of duality and reciprocity not only to extend already obtained results to a larger region but also to strengthen themselves substantially. Among other things we are able to settle the rationality and finiteness conjectures posed in the previous article.
We define an absolutely convergent series for the upper incomplete Gamma function $Γ(s,z)$ for $z\geq 1$ and $s\in \mathbb{C}$. We express this series using certain polynomials which we define using the Stirling numbers of the first kind. We prove that these polynomials have positive coefficients by defining a three-parameter family of integers and certain linear operators on vector spaces of polynomials. We then apply this series to obtain a formula for the Riemann xi function valid at any $s \in \mathbb{C}$.
Euler's gamma function is logarithmically convex on positive semi-axis. Additivity of logarithmic convexity implies that the function sum of gammas with non-negative coefficients is also log-convex. In this paper we investigate the series in reciprocal gamma functions, where each term is clearly log-concave. Log-concavity is not preserved by addition, so that non-negativity of the coefficients is now insufficient to draw any conclusions about the sum. We demonstrate that the sum is log-concave if the sequence of coefficients times factorial is log-concave and the sum is discrete Wright log-con…
Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of $q$-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions…
In this paper we determine the Fourier series expansion of the log-Barnes function. This is the analogue of the classical result of Kummer and Malmsten. Applying this expansion we get some integrals similar to the Espinosa-Moll log-Gamma integrals with respect to $\log G$. During the course of the paper some interesting log-Gamma integrals and a log-harmonic sum are evaluated.
In this paper, we present and prove some generalizations of some inequalities for the $p$-Gamma, $q$-Gamma and $k$-Gamma functions. Our approach makes use of the series representations of the psi, $p$-psi, $q$-psi and $k$-psi functions.
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ ζ(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $ζ(z)$, the generalized Stieltjes constants $γ_k(q)$, the Euler constant $γ$, Euler's gamma function $Γ(q)$ and the digamma function $ψ(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Eul…
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
A series transformation idea inspired by a formula of R. W. Gosper and some asymptotic expansions for the central binomial coefficients leads us to new accurate approximations for the Gamma function.
In the paper, the authors review origins, motivations, and generalizations of a series of inequalities involving several exponential functions and sums, establish three new inequalities involving finite exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers, survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finite gamma functions, present complete monotonicity of a linear combination of finite trigamma functions, construct a new ratio of finite…
We study analytic and arithmetic properties of the elliptic gamma function $$ \prod_{m,n=0}^\infty\frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \qquad |q|,|p|<1, $$ in the regime $p=q$; in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eistenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.
We discuss the best methods available for computing the gamma function $Γ(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function $\log Γ(z)$, the digamma function $ψ(z)$, and derivatives $Γ^{(n)}(z)$ and $ψ^{(n)}(z)$. Besides attempting to summarize the existing state of the art, we present some new formulas, estimates, bounds and algorithmic improvements and discuss implementation r…
The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.
In this paper, modified gamma and beta functions containing generalized M-series in their kernel are defined. Also, modified Gauss and confluent hypergeometric functions are defined using the modified beta function. Then, some properties of these modified special functions are obtained. Finally, the relationships between generalized special functions in the literature with newly defined modified special functions are examined.
An algorithm for computing the incomplete gamma function $γ^*(a,z)$ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincaré-type expansions, uniform asymptotic expansions and recurrence relations, depending on the parameter region. A relative accuracy $\sim 10^{-13}$ in the parameter region $(a,z) \in [-500,\,500] \times [-500,\,0)$ can be obtained when computing the function $γ^*(a,z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.
In this paper we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function $Γ\left( { - a,λa} \right)$ with large $a$ and fixed positive $λ$, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptot…
In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.
Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=γ(a,x)/Γ(a)$ and $Q(a,x)=Γ(a,x)/Γ(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability. The analytical approach from earlier literature is summarized and new initial estimates are derived for starting the inversion algorithms. The performance of the associated …
A doubly infinite set of series expansion for $1/π$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by an alternative computation of the Wronskian of the modified Bessel functions. The same formal expansion allows to discover also a new binomial identity.
We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,\infty)$ and the Zeta function on $(1,\infty)$.
A unique analytic solution of the generalized Dixon nonhomogeneous integral equation is derived in $C^{1}[0,A],\ A > 0$. It is written in terms of the Neumann series, which is expressed as a double series of residues at multiple poles of powers of the gamma-function.
In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series $F(a,b;c;x)$; we show that values of $F(a,b;c;x)$ at some points $x$ can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of $F(a,b;c;x)$ that can be obtained with this method. We find that this set includes almost all previously known value…
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second ki…
