scholarly journals A sequence involving an extended Struve function via a differential operator

2021 ◽  
Vol 39 (6) ◽  
pp. 129-137
Author(s):  
Gurmej Singh ◽  
Praveen Agarwal ◽  
Junesang Choi
Keyword(s):  
Differential Operator ◽  
Gamma Function ◽  
Summation Formula ◽  
Generating Relations ◽  
Sequence Of Functions

Various extensions of the Struve function have been presented and investigated. Here we aim to introduce an extended Struve function involving the $\mathtt{k}$-gamma function. Then, by using a known differential operator, we introduce a sequence of functions associated with the above introduced extended Struve function and investigate its properties such as generating relations and a finite summation formula. The results presented here, being very general, are also pointed out to yield a number of relatively simple identities.

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 A certain sequence of functions involving the Aleph function

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 187-191 ◽  
Author(s):  
P. Agarwal ◽  
M. Chand ◽  
İ. Onur Kiymaz ◽  
A. Çetinkaya
Keyword(s):  
Approximation Theory ◽  
Summation Formulas ◽  
Generating Relations ◽  
Sequence Of Functions

AbstractSequences of functions play an important role in approximation theory. In this paper, we aim to establish a (presumably new) sequence of functions involving the Aleph function by using operational techniques. Some generating relations and finite summation formulas of the sequence presented here are also considered.

scholarly journals Series expansion of the Gamma function and its reciprocal

2021 ◽  
Vol 27 (4) ◽  
pp. 104-115
Author(s):  
Ioana Petkova ◽  
Keyword(s):  
Differential Operator ◽  
Explicit Form ◽  
Series Expansion ◽  
Exponential Function ◽  
Gamma Function ◽  
Linear Functional ◽  
Maclaurin Series

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

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

scholarly journals Some generating-function equivalences

1975 ◽  
Vol 16 (1) ◽  
pp. 34-39 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Generating Relations ◽  
Sequence Of Functions

A generalization is given of a theorem of F. Brafman [1] on the equivalence of generating relations for a certain sequence of functions. The main result, contained in Theorem 2 below, may be applied to several special functions including the classical orthogonal polynomials such as Hermite, Jacobi (and, of course, Legendre and ultraspherical), and Laguerre polynomials.

ON THE INCOMPLETE GAMMA FUNCTION AND ITS NEUTRIX CONVOLUTION FOR NEGATIVE INTEGERS

2019 ◽  
Vol 10 (1) ◽  
pp. 30-51
Author(s):  
Mongkolsery Lin ◽  
◽  
Brian Fisher ◽  
Somsak Orankitjaroen ◽  
◽  
...  
Keyword(s):  
Gamma Function ◽  
Incomplete Gamma Function

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

On an extension of the Hardy-Hilbert theorem

2005 ◽  
Vol 42 (1) ◽  
pp. 21-35 ◽  
Author(s):  
J. Weijian ◽  
G. Mingzhe ◽  
G. Xuemei
Keyword(s):  
Beta Function ◽  
Summation Formula ◽  
Hilbert’S Inequality ◽  
Two Parameters

A weighted Hardy-Hilbert’s inequality with the parameter λ of form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\frac{{a_m b_n }}{{(m + n)^\lambda }}} < B^* (\lambda )\left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } a_{a_n }^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } b_n^q } } \right)^q }$$ \end{document} is established by introducing two parameters s and λ, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{p} + \tfrac{1}{q} = 1,p \geqq q > 1,1 - \tfrac{q}{p} < \lambda \leqq 2,B^* (\lambda ) = B(\lambda - (1 - \tfrac{{2 - \lambda }}{p}),1 - \tfrac{{2 - \lambda }}{p})$$ \end{document} is the beta function. B *(λ) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.

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