Vivek Sahai, Ashish Verma
In this paper we derive the infinite summation formulas of Srivastava's general triple hypergeometric function. Certain particular cases leading to infinite summation formulas for fourteen Lauricella and three Srivastavaś triple hypergeometric functions are also presented.
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The paper proposes to introduce incomplete Srivastava's triple hypergeometric matrix functions through application of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, reduction formula, recursion formula, recurrence relation and differentiation formula of the incomplete Srivastava's triple hypergeometric matrix functions.
We present here the $q$-analogues of certain transformations or reduction formulae for Srivastava-Daoust type double hypergeometric series. These reduction formulae are derived by utilizing the extended Bailey's Transform developed and studied by Joshi and Vyas [Int. J. Math. Sci., (12), 2005, 1909-1927]. A number of well-known $q$-hypergeometric transformations are also obtained as special cases of our results.
In this paper, we obtain a $(p,ν)$-extension of Srivastava's triple hypergeometric function $H_B(\cdot)$, together by using the extended Beta function $B_{p,ν}(x,y)$ introduced in arXiv:1502.06200. We give some of the main properties of this extended function, which include several integral representations involving Exton's hypergeometric function, the Mellin transform, a differential formula, recursion formulas and a bounded inequality.
Derivatives of Srivastava's hypergeometric functions with respect to their parameters
本文研究了Srivastava三重超几何函数对其参数的导数，推导了显式公式和多种微分关系，扩展了经典和多变量超几何函数的微分公式，为数学物理和工程应用提供了新工具。
Motivated by the work on hypergeometric summation theorems (recorded in the table III of Prudnikov et al. pp. 541-546), we have established some new summation theorems for Clausen's hypergeometric functions with unit argument in terms of $π$ and natural logarithms of some rational and irrational numbers. Also, we have given some modified summation theorems for Clausen's hypergeometric functions.
A summation formula is derived for the hypergeometric series of unit argument ${}_3F_2(1,1,c;d,n+2;1)$, where $n=0, 1, 2, \ldots$ and $\Re (d-c+n)>0$.
We consider the q-derivatives of the Srivastava and Daoust basic multivariable hypergeometric function with respect to the parameters. This function embodies a entire number of various q-hypergeometric series of one and several variables. Explicit equations are given for general case of summation indexes with positive real coefficients. As an example derivatives of q-analog of non-confluent Horn type hypergeometric function $H_3$ is presented.
After reviewing some fundamental facts from the theory of theta hypergeometric series we derive, using indefinite summation, several summation, transformation, and expansion formulas for multibasic theta hypergeometric series. Some of the identities presented here generalize corresponding formulas given in Chapter 11 of the Gasper and Rahman book [Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics And Its Applications 96, Cambridge University Press, Cambridge, 2004].
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational approximations of irrational numbers, we are able to obtain some necessary conditions of arithmetic flavor for a given hypergeometric sum to admit a gamma product formula. This kind of research seems to be new even in the most classical case of the Gauss hypergeometric series.
In this paper, we obtain analytical solutions of Laplace transform based some generalized class of the hyperbolic integrals in terms of hypergeometric functions ${}_3F_2 (\pm1)$, ${}_4F_3 (\pm1)$, ${}_5F_4(\pm1)$, ${}_6F_5(\pm1)$, ${}_7F_6(\pm1)$ and ${}_8F_7(\pm1)$ with suitable convergence conditions, by using some algebraic properties of Pochhammer symbols. In addition, reduction formulas for ${}_4F_3(1)$, ${}_7F_6(-1)$ and some new summation theorems (not recorded earlier in the literature of hypergeometric functions) for ${}_3F_2(-1)$, ${}_6F_5(\pm1)$, ${}_7F_6(\pm1)$ and ${}_8F_7(\pm1)$ …
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.
We use Andrews' $q$-analogues of Watson's and Whipple's $_3F_2$ summation theorems to deduce two formulas for products of specific basic hypergeometric functions. These constitute $q$-analogues of corresponding product formulas for ordinary hypergeometric functions given by Bailey. The first formula was obtained earlier by Jain and Srivastava by a different method.
In this paper, we study the solutions of the system of bilateral type matrix differential equations and presented these solutions in terms of Lauricella hypergeometric matrix functions of several variables and Srivastava's triple hypergeometric matrix functions of three variables. We also discuss the region of convergence and integral representation for these matrix functions.
Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As applications, we obtain a new $A_r$ elliptic Jackson summation, as well as several quadratic, cubic and quartic summation formulas.
This article describes the REDUCE package ZEILBERG implemented by Gregor Stölting and the author. The REDUCE package ZEILBERG is a careful implementation of the Gosper and Zeilberger algorithms for indefinite, and definite summation of hypergeometric terms, respectively. An expression $a_k$ is called a {\sl hypergeometric term} (or {\sl closed form}), if $a_{k}/a_{k-1}$ is a rational function with respect to $k$. Typical hypergeometric terms are ratios of products of powers, factorials, $Γ$ function terms, binomial coefficients, and shifted factorials (Pochhammer symbols) that are integer-line…
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further, using our truncated summation theorems, we obtain the Mellin transforms of the product of exponential function and Goursat's truncated hypergeometric function.
The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's $_{9}F_8(1)$ to $_{9}F_8(1)$ transformation are extended. These extensions are derived by applying the well-known Bailey's transform method along with the classical very well-poised summation and transformation theorems for very well-poised hypergeometric functions and the Rakha and Rathie's extension of the Saalschütz's theorem. To show importance and applications …
On the application of a hypergeometric identity to generate generalized hypergeometric reduction formulas
本文通过新的超几何恒等式推导出新的求和公式，并验证了相关函数的递推关系
Transformations and summations for bilateral basic hypergeometric series
本文研究双侧基本超几何级数的变换与求和，基于Zhang和Zhang、Wei和Yu的发现，推导出新的二次和三次求和公式及低级变换。
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan's 1-psi-1 summation and a very-well-poised 4-psi-6 summation as special cases.
Several new identities for elliptic hypergeometric series are proved. Remarkably, some of these are elliptic analogues of identities for basic hypergeometric series that are balanced but not very-well-poised.
We establish several summation formulae for hypergeometric and basic hypergeometric series involving noncommutative parameters and argument. These results were inspired by a recent paper of J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138-8141].
We give summation formulae for the bilateral basic hypergeometric series ${}_1ψ_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.
The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-fun…
Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials
本文通过阿斯凯-威尔逊多项式评估，推导出基本超几何级数的乘积公式，提出新的终止平衡${}_4ϕ_3$求和公式，并给出新的基本超几何变换和积分表示。
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $Γ$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $Γ…
Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by means of computer algebra. On the other hand, sometimes closed-forms of such extensions can be derived by induction. It is expected that the method used to obtain the different recursive equations can be applied to extend other hypergeometric summation formulas given in the literature.
In this paper, we obtain recursion formulas for the Kampé de Feŕiet hypergeometric matrix function. We also give finite and infinite summation formulas for the Kampé de Feŕiet hypergeometric matrix function.
The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of Meijer's $G$ function. For instance, we recover two- and three-term Thomae relations for ${}_3F_2$, give two- and three-term transformations for ${}_4F_3$ with one unit shift and ${}_5F_4$ with two unit shifts in the parameters, establish multi-term identities for general ${}_{p}F_{p-1}$ and several transformations for terminating Kampé de Fériet and Srivastava…
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.
We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
A summation formula is derived for the sum of the first m+1 terms of the 3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This summation formula is used to derive a formula for the sum of a terminating double hypergeometric series that arose in another project by one of us (C.D.)
On summation formulas in probability theory
本文提出若干在概率论中具有实用价值的求和公式，其证明依赖于分析学中近期应用于基本超几何函数的结果。
New duality transformation formulas are proposed for multiple elliptic hypergeometric series of type $BC$ and of type $C$. Various transformation and summation formulas are derived as special cases to recover some previously known results.
In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Bas…
Using multiple q-integrals and a determinant evaluation, we establish a multivariable extension of Bailey's nonterminating 10-phi-9 transformation. From this result, we deduce new multivariable terminating 10-phi-9 transformations, 8-phi-7 summations and other identities. We also use similar methods to derive new multivariable 1-psi-1 summations. Some of our results are extended to the case of elliptic hypergeometric series.
Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other hand, when more complex expressions arise, the latter function is not capable of representing them. The H-function is an alternative to overcome this issue, as it is a generalization of the Meijer-G function. In the present paper, a new identity for the H-function is derived. In short, this result enables one to split a particular H-function into the sum of…
We study multivariable (bilateral) basic hypergeometric series associated with (type $A$) Macdonald polynomials. We derive several transformation and summation properties for such series including analogues of Heine's ${}_2ϕ_1$ transformation, the $q$-Pfaff-Kummer and Euler transformations, the $q$-Saalschütz summation formula and Sear's transformation for terminating, balanced ${}_4ϕ_3$ series. For bilateral series, we rederive Kaneko's analogue of the ${}_1ψ_1$ summation formula and give multivariable extensions of Bailey's ${}_2ψ_2$ transformations.
Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically investigate several properties of each of these extended functions, namely their various integral representations, Mellin transforms, derivatives, transformations, summation formulas and asymptotics. Relevant connections of certain special cases of the main results presented here are also pointed out.
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. We also give a generating function of the sequence $\frac{(a)_n (1-a)_n}{(n!)^2}H_n$ as a combination of Gauss hypergeometric function and elementary functions.
We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.
Derivatives of Humbert confluent hypergeometric functions with respect to their parameters
本文研究了七种经典Humbert函数关于其参数的导数，推导出显式公式并给出闭合微分和约简公式，同时展示了Taylor型参数展开的应用。
In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic and the degenerate Miller-Paris transformations to get several new transformation and summation formulas for the generalized hypergeometric functions at a fixed argument. We further present an alternative approach for reducing the right hand sides resulting from our method to a single hypergeometric function which does not require the use of summation formul…
In the last decades, the theory of digamma function has been developed with a high impact of interest by many authors. Here, we established some interesting results for digamma function, and also we have computed the values of digamma function for positive integers, using the concept of hypergeometric series. An attempt has been made to present some summation theorems for Clausen's hypergeometric function. Results presented here are potentially useful in the further study of digamma function.
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6ψ_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts to an extension corresponding to the case of a nonreduced root system of certain summation identities associated to the reduced root systems that were recently conjectured by Aomoto and Ito and proved by Macdonald. By truncation, we obtain multidimensional analogues of the very-well-poised unilateral (basic) hypergeometric Rogers ${}_6ϕ_5$ and Dougall ${}_5F_4…
An elementary proof is given for a nonterminating "strange" cubic $_7F_6$-series summation formula of Gasper and Rahman, through the modified Abel lemma on summation by parts. As a byproduct, an interesting nonterminating $_3F_2({3}/{4})$-series identity conjectured by Gosper, is also rederived.
Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials
本文详尽探讨了Askey--Wilson多项式的$q$和$q^{-1}$对称子族的终止基本超几何表示与变换公式，并据此研究了其终止变换结构及$q$-Askey方案的对称群结构。
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
Reduction formulas for sums of products of hypergeometric functions can be traced back to Euler. This topic has an intimate connection to summation and transformation formulas, contiguous relations and algebraic properties of the (generalized) hypergeometric differential equation. Over recent several years, important discoveries have been made in this subject by Gorelov, Ebisu, Beukers and Jouhet and Feng, Kuznetsov and Yang. In this paper, we give a generalization of Feng, Kuznetsov and Yang identity covering also Ebisu's and Gorelov's formulas as particular cases.
Euler's transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as corollary a summation formula of Slater. From this formula new one-term evaluations of 2F1(-1) and 3F2(1) are derived, by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of 2F1(-1) with linearly…
We prove a master theorem for hypergeometric functions of Karlsson-Minton type, stating that a very general multilateral U(n) Karlsson-Minton type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson-Minton summation formula and many of its known generalizations as special cases, and it also implies several "Bailey-type" identities for U(n) hypergeometric series, including multivariable 10-W-9 transformations of Milne and Newcomb and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of mult…
Using Krattenthaler's operator method, we give a new proof of Warnaar's recent elliptic extension of Krattenthaler's matrix inversion. Further, using a theta function identity closely related to Warnaar's inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson-Minton-type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for el…
The double hypergeometric Kampé de Fériet series $F^{0:3}_{1:1}(1,1)$ depends upon 9 complex parameters. We present three cases with 2 relations between those 9 parameters, and show that under these circumstances $F^{0:3}_{1:1}(1,1)$ can be written as a ${}_4F_3(1)$ series. Some limiting cases of these transformation formulas give rise to new summation results for special $F^{0:3}_{1:1}(1,1)$'s. The actual transformation results arose out of the study of 9-$j$ coefficients.
On Bibasic Humbert hypergeometric function $Φ_1$
本文利用 $q$-微积分推导了双基 Humbert 超几何函数 $Φ_1$ 的 $q$-递推关系、$q$-偏导数关系、求和公式及新展开公式。
The main aim of the present work is to give some interesting the $q$-analogues of various $q$-recurrence relations, $q$-recursion formulas, $q$-partial derivative relations, $q$-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions $Ψ_{1}$ and $Ψ_{2}$ on two independent bases $q$ and $p$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types $Ψ_{1}$ a…
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
The beta integral method proved itself as a simple nonetheless powerful method of generating hypergeometric identities at a fixed argument. In this paper we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By application of our method to known transformation formulas we derive about forty hypergeometric identities, majority of which are believed to be new. We further apply some of these transformations to obtain several new summation formulas.
An extension of basic Humbert hypergeometric functions Φ1, Φ2 and Φ3
本文系统研究了基本Humbert合流超几何函数Φ1、Φ2和Φ3在参数约束下的q-偏导数公式、递推关系、偏微分方程、求和与变换公式及q-积分表示，推广了该领域的已知结果。
Factorization of Basic Hypergeometric Series
本文研究基本超几何级数的因式分解问题，重点分析一般 $_2ψ_2$ 级数，并建立与根系上基本超几何级数理论的联系，同时给出若干经典求和与变换公式的新证明。
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