Congruences concerning truncated hypergeometric series

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series(formula here)for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as(formula here)and(formula here)where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:(formula here)The result (1·6) is of particular interest because the argument and value of the F function are both rational.

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GAUSS SUMMATION AND RAMANUJAN-TYPE SERIES FOR 1/π

International Journal of Number Theory ◽

10.1142/s1793042112500169 ◽

2012 ◽

Vol 08 (02) ◽

pp. 289-297 ◽

Cited By ~ 11

Author(s):

ZHI-GUO LIU

Keyword(s):

Series Expansion ◽

Gamma Function ◽

Hypergeometric Series ◽

Summation Formula ◽

Expansion Formula ◽

Type Series

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan-type series for 1/π.

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An extension of Pochhammer’s symbol and its application to hypergeometric functions

Filomat ◽

10.2298/fil1702207m ◽

2017 ◽

Vol 31 (2) ◽

pp. 207-215 ◽

Cited By ~ 2

Author(s):

Mohammad Masjed-Jamei ◽

Gradimir Milovanovic

Keyword(s):

Gamma Function ◽

Hypergeometric Functions ◽

Hypergeometric Series ◽

Special Property ◽

Convergence Problem ◽

Pochhammer Symbol ◽

Generalized Hypergeometric Series ◽

General Properties

By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.

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Special values of the hypergeometric series

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100069723 ◽

1991 ◽

Vol 109 (2) ◽

pp. 257-261 ◽

Cited By ~ 6

Author(s):

G. S. Joyce ◽

I. J. Zucker

Keyword(s):

Gamma Function ◽

Hypergeometric Series ◽

Special Values ◽

Image Position

Recently, several authors [1, 3, 9] have investigated the algebraic and transcendental values of the hypergeometric seriesfor rational parameters a, b, c and algebraic arguments z. This work has led to some interesting new identities such asand where Γ(x) denotes the gamma function.

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Supercongruences for truncated hypergeometric series and p-adic gamma function

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000609 ◽

2018 ◽

Vol 168 (1) ◽

pp. 171-195 ◽

Cited By ~ 2

Author(s):

RUPAM BARMAN ◽

NEELAM SAIKIA

Keyword(s):

Finite Field ◽

Gamma Function ◽

Hypergeometric Series

AbstractWe prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

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Certain transformations for hypergeometric series in the p-adic setting

International Journal of Number Theory ◽

10.1142/s1793042115500359 ◽

2015 ◽

Vol 11 (02) ◽

pp. 645-660 ◽

Cited By ~ 2

Author(s):

Rupam Barman ◽

Neelam Saikia

Keyword(s):

Elliptic Curves ◽

Finite Fields ◽

Gamma Function ◽

Hypergeometric Functions ◽

Hypergeometric Series ◽

Transformation Formulas ◽

Trace Of Frobenius

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

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