The indefinite integral $$ \int x^αe^{ηx^β}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; λx^γ)dx, $$ where $α, η, β, λ, γ\ne0$ are real or complex constants and $_pF_q$ is the generalized hypergeometric function, is evaluated in terms of an infinite series involving the generalized hypergeometric function. Related integrals in which the exponential function $e^{ηx^β}$ is either replaced by the hyperbolic function $\cosh\left(ηx^β\right)$ or $\sinh\left(ηx^β\right)$, or the sinusoidal function $\cos\left(ηx^β\right)$ or $\sin\left(ηx^β\right)$, are also evaluated in terms of infinite series involving the generalized hypergeometric function $_pF_q$. Some application examples from applied analysis, in which some new Fourier and Laplace integrals (or transforms) are evaluated, are given. The analytical solution of the Orr-Sommerfeld equation (with a linear mean flow background) in the short-wave limit is expressed in terms of some infinite series involving the hypergeometric series $_2F_3$. Making use of the hyperbolic and Euler identities, some interesting series identities involving exponential, hyperbolic, trigonometric functions and the generalized hypergeometric function are also derived.
A formula for the non-elementary integral $\int e^{λx^α} dx$ where $α$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to $α= 2$, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function $_1F_1$ and another one in terms of the hypergeometric function $_1F_2$, are obtained for each …
Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain special cases of the main results presented here are also pointed out for the extended Gauss' hypergeometric and confluent hypergeometric functions.
We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-Konig and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions.
From the algebraic solution of $x^{n}-x+t=0$ for $n=2,3,4$ and the corresponding solution in terms of hypergeometric functions, we obtain a set of reduction formulas for hypergeometric functions. By differentiation and integration of these results, and applying other known reduction formulas of hypergeometric functions, we derive new reduction formulas of special functions as well as the calculation of some infinite integrals in terms of elementary functions.
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.
Recently, Rathie and Kılıçman (2014) employed Kummer-type transformation for $_{2}F_{2}(a, d+1; b, d; x)$ to develop certain classes of expansions theorems for $_{2}F_{2}(x)$ hypergeometric polynomial. Our aim is to deduce Kummer-type transformation for $_{2}F_{2}(a, d+2; b, d; x)$ and utilize it to develop some new expansion theorems for the confluent hypergeometric $_{2}F_{2}(x)$ polynomial. We also obtain a well-known result given by Kim et al. (Integral Transforms Spec. Funct. 23(6); 435-444, 2012) and many other new results as particular cases of our theorems.
For $a,b,p\in \mathbb{R}$, $-c\notin \mathbb{N\cup }\left\{ 0\right\} $ and $ θ\in \left[ -1,1\right] $, let \begin{equation*} U_{θ}\left( x\right) =\left( 1-θx\right) ^{p}F\left( a,b;c;x\right) =\sum_{n=0}^{\infty }u_{n}\left( θ\right) x^{n}. \end{equation*}% In this paper, we prove that the coefficients $u_{n}\left( θ\right) $ for $n\geq 0$ satisfies a 3-order recurrence relation. In particular, $ u_{n}\left( 1\right) $ satisfies a 2-order recurrence relation. These offer a new way to study for hypergeometric function. As an example, we present the necessary and sufficient conditions such th…
We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group.
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 provide a short and elementary proof of the main technical result of the recent article "On the uniqueness of Clifford torus with prescribed isoperimetric ratio" by Thomas Yu and Jingmin Chen. The key of the new proof is an explicit expression of the central function (${\sf Iso}$, to be proved bijective) as a quotient of Gaussian hypergeometric functions.
The authors establish the necessary and sufficient conditions under which certain combinations of Gaussian hypergeometric function and elementary function are monotone in the parameter, which generalize the recent results of generalized elliptic integrals of the first and second kinds obtained by Qiu et al. Moreover, the authors also prove two monotonicity theorems of generalized elliptic integrals from another point of view.
Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are consid…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show in this paper the surprising fact that this integral does not depend on the parameters of the hypergeometric functions.
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…
In this paper, the product of parabolic cylinder functions $D_ν(\pm z)D_{ν+μ-1}(z)$, with different parameters $μ$ and $ν$, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder functions.
Just as with the Gauss hypergeometric function, particular cases of the local Heun function can be Liouvillian (that is, "elementary") functions. One way to obtain these functions is by pull-back transformations of Gauss hypergeometric equations with Liouvillian solutions. This paper presents the Liouvillian solutions of Heun's equations that are pull-backs of the parametric hypergeometric equations with cyclic or dihedral monodromy groups.
Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant $γ$. The proof is by reduction to known irrationality criteria for $γ$ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, $γ$, and logarithms from Nesterenko-type series of rational functions. In the Appendix, Sergey Zlobin gives a change-of-variables proof that the series and the double integral are equal.
We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities.
Identities involving finite sums of products of hypergeometric functions and their duals have been studied since 1930s. Recently Beukers and Jouhet have used an algebraic approach to derive a very general family of duality relations. In this paper we provide an alternative way of obtaining such results. Our method is very simple and it is based on the non-local derangement identity.
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series.
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.
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. …
Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and then put forward some fundamental results in the paper. Next, we will derive some properties of certain functions like extended Gauss hypergeometric functions, extended Appell's hypergeometric functions $\mathbf{F}_{1}^{(κ_{l})}$, $\mathbf{F}_{2}^{(κ_{l})}$, and extended Lauricella's hypergeometric function $\mathbf{F}_{D,(κ_{l})}^{(r)}$,$\mathbf{F}_{A,(κ_{…
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.
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.
The non-elementary integrals $\text{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α\leβ+1$ and $\text{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α\le2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$, and their asymptotic expressions for $|x|\gg1$ are also derived. The integrals of the form $\int [\sin^n{(λx^β)}/(λx^α)] dx$ and $\int [\cos^n{(λx^β)}/(λx^α)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$, and then evaluated. $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$ are also…
Evaluation of terminating and non-terminating sums containing the digamma function
Shanghai Jiao Tong University（上海交通大学）；Keldysh Institute of Applied Mathematics of Russian Academy of Sciences（俄罗斯科学院应用数学凯尔迪什研究所）；Holon Institute of Technology（霍隆理工学院）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Bennett University（贝内特大学）
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(α)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and $α$ small or large, uniformly for unbounded real and complex values of $x$. The new expansions extend the range of computability of $L_n^{(α)}(x)$ compared to previous expansions, in particular with respect to higher terms and large values of $α$. Numerical evidence of their accuracy for real and complex values of $x$ is provided.
We present a new `elementary' proof of the irrationality of $ζ(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.
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…
The main object of this article is to present an interesting double integral involving generalized Bessel-Maitland function defined by Ghaysuddin et al., which is expressed in terms of generalized (Wright) hypergeometric function. We also considered some special cases as an application of the main result.
In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters exceeds one of the bottom parameters by a positive integer or reversely one of the bottom parameters exceeds one of the top parameters by a positive integer or both. We show that all such cases reduce to the case of the unit parameter difference. The latter case, in turn, can be expressed in terms of certain linear combination of two series involving the loga…
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
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].
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…
We evaluate some integrals over $[0,\infty)$ of quotients of powers of the hyperbolic functions $\sinh x$ and $\cosh x$ using a hypergeometric approach. Some of these results appear to be new but several verify the entries in the table of integrals of Gradshteyn and Rhyzik.
The Kampé de Fériet double series $F_{1:1;1}^{1:1;1}$ is studied through the solution to the associated first-order nonhomogeneous differential equation. It is shown that the integral of $t^{β+l}M(\cdot;β;λt)M(\cdot;β;-λt)$ over $t\in[0,T]$, $T\geq0$, $l=0,1,\ldots$, $\Reβ+l>-1$, is a linear combination of functions $F_{1:1;1}^{1:1;1}$. The integral is a generalization of a class of so-called Coulomb integrals involving regular Coulomb wave functions.
Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, some relations following from the decomposition formula for one multidimensional Lauricell hypergeometric function are determined.
We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric Series" book (which we will refer to as BHS), the Bailey [1935] and Slater [1966] books, and in some papers; thus providing deeper insights into the theory of q-series. Our main emphasis is on methods that can be used to derive formulas, rather than to just verify previously derived or conjectured formulas. In section 5 this approach leads to the derivation of a…
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.
In these lecture notes I give an elementary introduction to elliptic hypergeometric functions. I focus on motivating the main ideas and constructions, rather than giving a comprehensive survey. The lectures include a brief explanation of the historical origin of elliptic hypergeometric functions in the context of solvable lattice models. In particular, I give a new proof of the fact that fused Boltzmann weights for the elliptic solid-on-solid model can be expressed as elliptic hypergeometric sums.
The non-elementary integrals $\mbox{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α>β+1$ and $\mbox{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α>2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx, β\ge1, α>β+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.
Various methods to obtain the analytic continuation near $z=1$ of the hypergeometric series $_{p+1}F_p(z)$ are reviewed together with some of the results. One approach is to establish a recurrence relation with respect to $p$ and then, after its repeated use, to resort to the well-known properties of the Gaussian hypergeometric series. Another approach is based on the properties of the underlying generalized hypergeometric differential equation: For the coefficients in the power series expansion around $z=1$ a general formula, valid for any $p$, is found in terms of a limit of a terminating Sa…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at $0$ and at $\infty$. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the $q$-derivative, and the related raising operator. Next we consider the $q$-hypergeometric operator in a special case, and we show that there is a natural…
We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for gamma, the beta integral, and an integral for the digamma function. (I thank P. Sebah, S. Zlobin, and W. Zudilin for valuable suggestions used in the paper.)
