Recursion formulas for multivariable hypergeometric functions

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX>. Each of these three triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function <TEX>$H_B$</TEX>.

In this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

In this paper, we obtain solutions of some multiple series identities involving bounded multiple sequences. We also derive hypergeometric forms of these identities involving Kampé de Fériet double hypergeometric function, Srivastava’s triple hypergeometric function.

In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known q-analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the q-analogues of Srivastava’s three triple hypergeometric functions in terms of ϕ34q-hypergeometric and q-Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of F34 hypergeometric and Kampé de Fériet functions.

Motivated mainly by certain interesting extensions of the <TEX>${\tau}$</TEX>-hypergeometric function defined by Virchenko et al. [11] and some <TEX>${\tau}$</TEX>-Appell's function introduced by Al-Shammery and Kalla [1], we introduce here the <TEX>${\tau}$</TEX>-Lauricella functions <TEX>$F_A^{(n),{\tau}_1,{\cdots},{\tau}_n}$</TEX>, <TEX>$F_B^{(n),{\tau}_1,{\cdots},{\tau}_n}$</TEX> and <TEX>$F_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$</TEX> and the confluent forms <TEX>${\Phi}_2^{(n),{\tau}_1,{\cdots},{\tau}_n}$</TEX> and <TEX>${\Phi}_D^{(n),{\tau}_1,{\cdots},{\tau}_n}$</TEX> of n variables. We then systematically investigate their various integral representations of each of these <TEX>${\tau}$</TEX>-Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.

On certain results related to the hypergeometric function FK

Functions NF (x1,...,xN) which are a straightforward generalisation of standard hypergeometric functions of N variables are introduced. A convenient operator representation is established for these functions which permits one to consider, in a uniform and simple way, standard hypergeometric functions (generalised hypergeometric series of one variable, Appell, Kampe de Feriet and Lauricella functions, etc.) as well as many other nameless hypergeometric series arising in physical and quantum chemical applications. Generating functions, generalisations of Bailey (1935) and Chaundy (1943) expansions, recurrence relations, reduction rules and some other topics are discussed.

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.

Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

Motivated by the recent investigations of several authors, in this paper, we derive several new expansion formulas involving a generalized Hurwitz-Lerch zeta function introduced and studied recently by Srivastavaet al. (Integral Transforms Spec. Funct. 22:487-506, 2011). These expansions are obtained by using some fractional calculus theorems such as the generalized Leibniz rules for the fractional derivatives and the Taylor-like expansions in terms of different functions. Several (known or new) special cases are also considered.MSC:11M25, 11M35, 26A33, 33C05, 33C60.

Recently, Srivastava and Attiya [H.M. Srivastava and A.A. Attiya, An integral operator associated with the Hurwitz–Lerch Zeta function and differential subordination, Integral Transforms Spec. Funct. 18 (2007), pp. 207–216] introduced a linear integral operator on the class of analytic function associated with Hurwitz–Lerch Zeta function. Using this Srivastava–Attiya operator and generalized Janowski functions, certain classes of analytic functions related with the functions of bounded boundary and bounded radius rotation are introduced and studied. Inclusion results, rate of growth of coefficients, radius problem and behaviour of these classes under certain integral operators are investigated.

The aim of this research paper is to obtain explicit expressions of 2F1 [a,b 1/2(a+b ? ?+1); 1+x/2] in the most general case for any ? = 0, 1, 2, ... For ? = 0, we have the well known, interesting and useful formula due to Kummer which was proved independently by Ramanujan. The results presented here are obtained with the help of known generalizations of Gauss?s second summation theorem for the series 2F1(1/2), which were given earlier by Rakha and Rathie [Integral Transforms Spec. Func. 22 (11) (2011), 823-840]. The results are further utilized to obtain new hypergeometric identities by using beta integral method developed by Krattenthaler &amp; Rao [J. Comput. Appl. Math. 160 (2003), 159-173]. Several interesting results due to Ramanujan, Choi, et. al. and Krattenthaler &amp; Rao follow special cases of our main findings.

This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Two different aerating transforms are introduced and closed-form expressions are derived for the Hankel transform of such aerated sequences. Combinations of both aerating and the Hankel transforms are also considered. Our results are general and can be applied to a wide class of integer sequences. As an application, we use our tools on the sequence of shifted Catalan numbers (Cn+t)n∈ℕ 0. For that purpose, we need to evaluate the Hankel and Hankel-like determinants based on the Catalan numbers. Our approach is based on the results of Gessel and Viennot [Determinants, paths, and plane partitions preprint. Available from: http://www.cs.brandeis.edu/ ira] and more recent results of Krattenthaler [Determinants of (generalised) Catalan numbers. J. Statist. Plann. Inference 2010; 240: 2260–2270]. We generalize a sequence obtained by the series reversion of Q(x)=x/(1+α x+β x 2) (studied in our previous paper [Bojičić R, Petković MD, Barry P. Hankel transform of a sequence obtained by series reversion. Integral Transforms Spec. Funct. DOI: 10.1080/10652469.2011.640326]) and provide the Hankel transform evaluation of that sequence and its shifted sequences.

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.