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Register a new userVarious considerations on hypergeometric series
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Leonhard Euler
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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 think they understand please I would appreciate it if you could email me. I am presently examining Eulers work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.
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We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newtons interpolation formula.
This is the translation of Leonhard Eulers paper De Seriebus divergentibus written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-text{etc.}$ and uses different methods to sum it. He finds a value of about $0.59...$.
We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.
A translation of Kummer`s paper On certain definite integrals and infinite series
We investigate a certain linear combination $K(vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V={(a,b,c,d,e,f,g)inBbb C^7colon e+f+g-a-b-c-d=1}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $mu_1,mu_2,mu_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(mu_1vec{x})$, $K(mu_2vec{x})$, and $K(mu_3vec{x})$, provided no two of the $mu_j$s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|atop 3})=({32atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(mu_1,mu_2,mu_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $mu_j$s, which in turn is defined in terms of the expression of the $mu_j$s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.
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