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.)
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.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several 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 introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.
In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler $Γ$-functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical evaluations, but the main inspiration is from the contiguity method recently applied by Akihito Ebisu.
We obtain addition formulas for $_{p}F_{p}$ and $_{p+1}F_{p}$ generalized hypergeometric functions with general parameters. These are utilized in conjunction with integral representations of these functions to derive Kummer- and Euler-type transformations that express $_{p}F_{p}\left(x\right)$ and $_{p+1}F_p\left(x\right)$ in the form of sums of $_{p}F_{p}\left(-x\right)$ and $_{p+1}F_p\left(-x\right)$ functions, respectively.
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 consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.
A new multiple-integral representation of a general family of very-well-poised hypergeometric series is proved. Inspite of an analytic character of the result, it is motivated by the recent arithmetic progress for the values of the Riemann zeta function at odd integers. The multiple integrals are of Euler-type and can be viewed as a natural generalization of Beukers' integrals for $ζ(2)$ and $ζ(3)$.
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed.
We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.
E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers th…
Inspired by certain interesting recent extensions of the gamma, beta and hypergeometric matrix functions, we introduce here new extension of the gamma and beta matrix function. We also introduce new extensions of the Gauss hypergeometric matrix function, confluent hypergeometric matrix function, Appell matrix function and Lauricella matrix function of three variables in terms of the new extended beta matrix function. Then we investigate certain properties of these extended matrix functions such as the integral representations, differential formulae and recurrence relations.
We introduce gamma structures on regular hypergeometric D--modules in dimension 1 as special one--parametric systems of solutions on the compact subtorus. We note that a balanced gamma product is in the Paley--Wiener class and show that the monodromy with respect to the gamma structure is expressed algebraically in terms of the hypergeometric exponents. We compute the hypergeometric monodromy explicitly in terms of certain diagonal matrices, Vandermonde matrices and their inverses (or generalizations of those in the resonant case).
The aim of the paper is to relate computational and arithmetic questions about Euler's constant $γ$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical formula for $γ$ due to Ramanujan, together with Vacca's and Gosper's series for $γ$, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler's constant. The main tools are Euler-type integrals and hypergeometric series.
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. …
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Laboratory of Theoretical Physics, JINR（约飞联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里安物理研究所）；National Research University Higher School of Economics（国立研究大学高等经济大学）
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler's and Selberg's beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for …
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.
We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second and third Euler angles in explicit and real forms by means of multiple hypergeometric functions (Lauricella, functions). Releasing the weight load but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle completing some previous treatments of…
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.
For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on …
We obtain two new Thomae-type transformations for hypergeometric series with r pairs of numeratorial and denominatorial parameters differing by positive integers. This is achieved by application of the so-called Beta integral method developed by Krattenthaler and Rao [Symposium on Symmetries in Science (ed. B. Gruber), Kluwer (2004)] to two recently obtained Euler-type transformations. Some special cases are given.
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.
Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily extended to higher order multiple gamma functions). Some applications of these Fourier series are explored.
Using contiguous relations we construct an infinite number of continued fraction expansions for ratios of generalized hypergeometric series 3F2(1). We establish exact error term estimates for their approximants and prove their rapid convergences. To do so we develop a discrete version of Laplace's method for hypergeometric series in addition to the use of ordinary (continuous) Laplace's method for Euler's hypergeometric integrals.
In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.
General elliptic hypergeometric functions are defined by elliptic hypergeometric integrals. They comprise the elliptic beta integral, elliptic analogues of the Euler-Gauss hypergeometric function and Selberg integral, as well as elliptic extensions of many other plain hypergeometric and $q$-hypergeometric constructions. In particular, the Bailey chain technique, used for proving Rogers-Ramanujan type identities, has been generalized to integrals. At the elliptic level it yields a solution of the Yang-Baxter equation as an integral operator with an elliptic hypergeometric kernel. We give a brie…
We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.
We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transfo…
Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system $C_n$. In a special $n=1$ case, the simplest $p\to 0$ limit is shown to lead to a new class of $q$-hypergeometric identities. Symmetries of a rarefied elliptic analogue of the Euler-Gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed. Some extensions of the latter function to $C_n$ and $A_n$ root system…
A special singular limit $ω_1/ω_2\to 1$ is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and the corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin-Barnes type integrals of particular Pochhammer symbol products.
In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4]. Some recurrence relations, transformation formulas, Mellin transform and integral representations are obtained for these generalizations. Further, an extension of the Riemann-Liouville fractional derivative operator is established.
We consider differences between $\log Γ(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-λ$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(λ)$, and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases $λ=0$ and $λ=1/2$, we recover results of Sonin, Nörlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of $1/x$ as a difference $F(x+1)-F(x)$, and a back…
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…
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 te…
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.
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）
In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the the generalized hypergeometric function over the branch cut, establish new inequalities for this function in the half plane Re(z)<1. Furthermore,…
We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between $S^1\times S^3/\mathbb{Z}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ supers…
In this paper, we prove that $ζ$ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ^{(n)}, Γ^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n\geq 1$ are positive integers. We extended the result that $ζ$ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ', Γ''$ over the field of complex numbers, which is proved by Li and Ye[7].
In this paper we study the integral of type \[_{δ,a}Γ_{ρ,b}(x) =Γ(δ,a;ρ,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^δ}{a}-\frac{t^{-ρ}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Func…
This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. It contains an outline of a general heuristic scheme for building univariate special functions through se…
We consider a special degeneration limit $ω_1\to - ω_2$ (or $b\to {\rm i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the ${\rm SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperb…
