Three Term Relations for the Hypergeometric Series

Ebisu, Akihito

doi:10.1619/fesi.55.255

Cited by 20 publications

(36 citation statements)

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“…II, §2.9, formulas (1)-(24)]. Ebisu [4,Lemma 2.2] showed that each of Kummer's solutions, say 2 K 1 (a; z), admits a three-term relation of the following form: for every integer vector p = (p, q; r) ∈ Z 3 , 2 K 1 (a + p; z) = ψ(a; p) r(a; z) 2 K 1 (a; z) + φ(a; p) q(a; z) 2…”

Section: Kummer's 24 Solutions and Ebisu Symmetriesmentioning

confidence: 99%

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Hypergeometric series with gamma product formula

Iwasaki

2017

Indagationes Mathematicae

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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.

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Section: Kummer's 24 Solutions and Ebisu Symmetriesmentioning

confidence: 99%

“…If the answer is "yes", we say that the rational solution λ essentially comes from contiguous relations. It is easy to see that formula (2) can be recovered from formula (10) via relation (8). Indeed, we have only to replace w by w/k in (10) and use the multiplication formula (9) in the other way round.…”

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confidence: 99%

Hypergeometric series with gamma product formula

Iwasaki

2017

Indagationes Mathematicae

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“…where p(∂) is an element of the ring of the differential operators in x over Q(a, b, c, x), and q(x), r(x) ∈ Q(a, b, c, x) (cf. (2.4) in [Eb1]).…”

Section: Contiguity Operatorsmentioning

confidence: 92%

Special Values of the Hypergeometric Series

Ebisu

2017

Memoirs of the AMS

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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 values and many previously unknown values.

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“…The equation ST 4 of order four that appeared in the forgoing sections is studied in Section 11. Though each of the equations E 6 , SE 6 , E 5 , SE 5 , E 4 , SE 4 and Z 4 has an accessory parameter, ST 4 not. This equation was first studied in [12].…”

Section: Introductionmentioning

confidence: 99%

Shift relations and reducibility of some Fuchsian differential equations of order 2,...,6 with three singular points

Ebisu¹,

Haraoka²,

Ochiai³

et al. 2021

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A Fuchsian differential equation of order six with nine free exponents as parameters and with three singular points is presented. This equation has various symmetries, which specify the accessory parameter as a polynomial of the local exponents. For some shifts of exponents, the shift operators are found, which lead to reducibility conditions of the equation. By several specializations of the parameters and successive factorizations of the equation, it produces several new equations and also known ones. For each equation, shift operators are studied, and when the equation is reducible, we observe the way of factorization and discuss the relation with the shift operators.

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Three Term Relations for the Hypergeometric Series

Abstract: Abstract. Three hypergeometric series F ða; b; c; xÞ with the same parameters ða; b; cÞ up to additive integers are linearly related over rational functions in x. This paper makes this linear relation explicit: the coe‰cients are given from sums of products of hypergeometric series.

Cited by 20 publications

References 1 publication

Hypergeometric series with gamma product formula

Hypergeometric series with gamma product formula

Special Values of the Hypergeometric Series

Shift relations and reducibility of some Fuchsian differential equations of order 2,...,6 with three singular points

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