GAUSS SUMMATION AND RAMANUJAN-TYPE SERIES FOR 1/π

2012 ◽  
Vol 08 (02) ◽  
pp. 289-297 ◽  
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.

scholarly journals Approximation of Euler Number Using Gamma Function

2015 ◽  
Vol 12 (3) ◽  
Author(s):  
Shekh Zahid ◽  
Prasanta Ray
Keyword(s):  
Series Expansion ◽  
Gamma Function ◽  
Taylor Series ◽  
Function Approximation ◽  
Euler Number ◽  
Irrational Number ◽  
Taylor Series Expansion ◽  
Summation Formula ◽  
Graphical Methods

This research presents a formula to calculate Euler number using gamma function. The representation is somewhat similar to Taylor series expansion of e. The number e is presented as sum of an integral and decimal part. But it is well known that e is an irrational number, and therefore such an expression for e does not exist in principle, that is why we are approximately representing it for use in computation. The approximation of our formula increases as we increase the value of index in the summation formula. We also analyse the approximation of our formula by both numerical and graphical methods. KEYWORDS: Euler Number, Gamma Function, Approximation, Computing, L’Hospital's Rule

Special values of the hypergeometric series II

2001 ◽  
Vol 131 (2) ◽  
pp. 309-319 ◽  
Author(s):  
I. J. ZUCKER ◽  
G. S. JOYCE
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Elliptic Integral ◽  
Singular Values ◽  
Complete Elliptic Integral ◽  
Special Values ◽  
Gaussian 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.

scholarly journals SIGMOIDAL-TYPE SERIES EXPANSION

2008 ◽  
Vol 49 (03) ◽  
pp. 431 ◽  
Author(s):  
BEONG IN YUN
Keyword(s):  
Series Expansion ◽  
Type Series

Congruences concerning truncated hypergeometric series

2017 ◽  
Vol 147 (3) ◽  
pp. 599-613 ◽  
Author(s):  
Bing He
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

On Jackson’s proof of Ramanujan’s 1ψ1 summation formula

2018 ◽  
Vol 14 (02) ◽  
pp. 313-328
Author(s):  
Jorge Luis Cimadevilla Villacorta
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Summation Formula ◽  
Ramanujan’S Formula

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

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

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 207-215 ◽  
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.

scholarly journals Transformation formulas for terminating Saalschützian hypergeometric series of unit argument

1995 ◽  
Vol 8 (2) ◽  
pp. 189-194
Author(s):  
Wolfgang Bühring
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Series ◽  
Summation Formula ◽  
Summation Formulas ◽  
Finite Sums ◽  
Transformation Formulas ◽  
Independent Parameters

Transformation formulas for terminating Saalschützian hypergeometric series of unit argument p+1Fp(1) are presented. They generalize the Saalschützian summation formula for 3F2(1). Formulas for p=3,4,5 are obtained explicitly, and a recurrence relation is proved by means of which the corresponding formulas can also be derived for larger p. The Gaussian summation formula can be derived from the Saalschützian formula by a limiting process, and the same is true for the corresponding generalized formulas. By comparison with generalized Gaussian summation formulas obtained earlier in a different way, two identities for finite sums involving terminating 3F2(1) series are found. They depend on four or six independent parameters, respectively.

Sign in / Sign up

Export Citation Format

Share Document