Rafael Diaz, Eddy Pariguan
We introduce the $k$-generalized gamma function $Γ_k$, beta function $B_k$, and Pochhammer $k$-symbol $(x)_{n,k}$. We prove several identities generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. We provided integral representation for the $Γ_k$ and $B_k$ functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we first introduce certain forms of extended incomplete Pochhammer symbols which are then used to define families of extended incomplete generalized hypergeometric functions. For these functions, we investigate various properties including the integral representations, derivative formula, certain generating function and fractional integrals (and derivatives) relationships. Some special cases of the main results are also deduced.
A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function
本文通过Jacobi多项式和广义超几何函数的展开，推导出涉及Pochhammer符号的新的恒等式和积分闭形式。
Continuous Analogues for the Pochhammer Symbol
本研究沿Díaz与Cano的组合对象连续化思路，提出Pochhammer符号的连续模拟，拓展了离散组合对象的连续对应理论。
In this paper, the incomplete Pochhammer ratios are defined in terms of the incomplete beta function $B_{y}(x,z)$. With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric and Appell's functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relation. Furthermore, an incomplete Riemann-Liouville fractional derivative operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hyp…
The method of brackets is a method of integration based upon a small number of heuristic rules. Some of these have been made rigorous. An example of an integral involving the Bessel function is used to motivate a new evaluation rule.
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.
Log-concavity and log-convexity of series containing multiple Pochhammer symbols
本文研究含多个 Pochhammer 符号的四类幂级数，通过广义 Turánian 系数的正负性，由一般序列的对数凹/凸性推出级数和的相应性质，并应用于广义超几何函数与 Fox-Wright 函数。
We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q|=1. In this framework, we prove the Heine's relations and a connection formula for the q-hypergeometric function of the Barnes type. We also prove an orthogonality relation of the q-little Jacobi polynomials at |q|=1.
In this paper we introduce the Two Parameter Gamma Function, Beta Function and Pochhammer Symbol. We named them, as p - k Gamma Function, p - k Beta Function and p - k Pochhammer Symbol and denoted as $_{p}Γ_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ respectively. We prove the several identities for $_{p}Γ_{k}(x), $ $_{p}B_{k}(x,y) $ and $_{p}(x)_{n,k} $ those satisfied by the classical Gamma, Beta and Pochhammer Symbol. Also we provide the integral representation for the $_{p}Γ_{k}(x) $ and $_{p}B_{k}(x,y) $.
We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Also, we compare two different differentiation formulas of the generalized hypergeometric function with respect to the parameters. For some particular cases, we recover some results found in the literature. Finally, all the results have been numerically checked.
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…
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 ${}_8F_7(\pm1)$ …
We introduce a natural method of computing antiderivatives of a large class of functions which stems from the observation that the series expansion of an antiderivative differs from the series expansion of the corresponding integrand by just two Pochhammer symbols. All antiderivatives are thus, in a sense, "hypergeometric". And hypergeometric functions are therefore the most natural functions to integrate. This paper would like to make two points: First, the method presented is easy. So much so that it can be taught in undergraduate university level. And second: It may be used to prove some of…
We are going to study properties of "hypergeometrization" -- an operator which act on analytic functions near the origin by inserting two Pochhammer symbols into their Taylor series. In essence, this operator maps elementary function into hypergeometric. The main goal is to produce number of "change of variable" formulas for this operator which, in turn, can be used to derive great number of transform for multivariate hypergeometric functions.
In this paper, we obtain the analytical solutions of Laplace transforms based some novel integrals with suitable convergence conditions, by using hypergeometric approach (some algebraic properties of Pochhammer symbol and classical summation theorems of hypergeometric series ${}_{2}F_{1}(1)$, ${}_{2}F_{1}(-1)$ , ${}_{4}F_{3}(-1)$) . Also, we obtain the Laplace transforms of arbitrary powers of some finite series containing hyperbolic sine and cosine functions having different arguments, in terms of hypergeometric and Beta functions. Moreover, Laplace transforms of even and odd positive integra…
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…
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。
We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values may be obtained. It may be emphasized that we make no reliance on the WZ method for hypergeometric summation. Additional …
In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine transforms, Infinite series decomposition identity and some algebraic properties of Pochhammer's symbol. Also we have given some generalizations of the Ramanujan's integral $\textbf{R}_{C}(m,n)$ in the form of integrals $\textbf{I}^{*}_{C}(\upsilon,b,c,λ,y), \textbf{J}_C (\upsilon,b,c,λ,y), \textbf{K}_{C} (\upsilon,b,c, λ,y), \textbf{I}_{C}(\upsilon,b,λ,y)$ and sol…
We show that the complex hypergeometric function describing $6j$-symbols for $SL(2,\mathbb{C})$ group is a special degeneration of the $V$-function -- an elliptic analogue of the Euler-Gauss $_2F_1$ hypergeometric function. For this function, we derive mixed difference-recurrence relations as limiting forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.
Our present investigation is mainly based on the $k$-hypergeometric functions which are constructed by making use of the Pochhammer $k$-symbol \cite{Diaz} which are one of the vital generalization of hypergeometric functions. We introduce $k$-analogues of $F_{2}\ $and $F_{3}$ Appell functions denoted by the symbols $F_{2,k}\ $and $F_{3,k}\ $respectively, just like Mubeen et al. did for $F_{1}$ in 2015 \cite{Mubeen6}. Meanwhile, we prove some main properties namely integral representations, transformation formulas and some reduction formulas which help us to have relations between not only $k$-…
In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, Kıymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin transform are obtained. Besides, the unified Bessel, modified Bessel, spherical Bessel and Bessel-Clifford fun…
The Wright function -- hypergeometric representation and symbolical evaluation
本文提出一种将 Wright 函数表示为超几何函数与多项式有限和的符号化算法，并在 Maxima 中实现，可用于验证 Wright 函数数值算法。
The main object of this work is to show how some rather elementary techniques based upon certain inverse pairs of symbolic operators would lead us easily to several decomposition formulas associated with confluent hypergeometric functions of two and more variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities several decomposition formulas are found, which express the aforementioned hypergeometric functions in terms of such simpler functions as the products of the Gauss hypergeometric function…
Expression of special stretched $9j$ coefficients in terms of $_5F_4$ hypergeometric series
本研究针对一般非平凡$9j$符号无法表示为$_5F_4$超几何级数的问题，利用Dougall-Ramanujan恒等式，证明了特殊拉伸$9j$符号可改写为$_5F_4(1)$超几何级数，拓展了角动量系数的级数表达形式。
By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Lauricella's hypergeometric functions $F_A^{(r)}, F_B^{(r)}, F_C^{(r)}$ and $F_D^{(r)}$ in $r$ variables. In the three-variable case some of these operational representations are constructed and applied in order to derive the corresponding decomposition formulas when $r = 3$ . With the help of these new inverse pairs of symbolic operators, a total 20 decomposition formulas and integral representations are found.
A new product formula for $(z;q)_\infty$, with applications to asymptotics
本文提出一个新的$(z;q)_\infty$乘积公式，并利用该公式研究$q$趋近于1时的渐近展开问题。
In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined. The obtained formulas of an analytic continuation express known hypergeometric Appell function $ F_2 ({a;b_1, b_2 ;c_1, c_2 ;x,y}) $ which theory is well studied.
We introduce a generalization of elliptic 6j-symbols, which can be interpreted as matrix elements for intertwiners between corepresentations of Felder's elliptic quantum group. For special parameter values, they can be expressed in terms of multivariable elliptic hypergeometric series related to the root system A_n. As a consequence, we obtain new biorthogonality relations for such series.
On the incomplete Lauricella matrix functions of several variables
本文通过不完全Pochhammer矩阵符号引入n变量的不完全Lauricella矩阵函数，推导其性质，并建立与其他特殊矩阵函数的联系。
A hypergeometric basis for the Alpert multiresolution analysis
本文构建了用于阿尔佩特多分辨率分析的分段超几何多项式正交基，通过超几何函数的傅里叶变换和矩阵方程的递推关系，推导出新的正交关系和恒等式。
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…
The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Moebius automorphisms and F-homotopies, and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of 3F2 type is examined, and Pfaff and Euler transformations of 3F2(a1,a2,e+1;b1,e;x) are derived. They ex…
In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types of identities for those functions. Some of these identities like differential equations, power series representations, and hypergeometric representations can even dealt with algorithmically, i.\ e.\ they can be computed by the Computer Algebra system, rather than only verified. The types of functions that can be treated by the given technique cover the genera…
Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function
本文从量子曲线视角研究非耦合BPS结构与拓扑递归的对应，证明超几何型谱曲线的Borel重求和Voros符号可求解Bridgeland的BPS Riemann-Hilbert问题，并揭示Bridgeland $τ$-函数与相应势函数及拓扑递归配分函数之间的内在联系。
We prove a formula expressing a general n by n Toeplitz determinant as a Fredholm determinant of an operator 1-K acting on l_2({n,n+1,...}), where the kernel K admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see math.RT/9907127, and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。