scholarly journals Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1952
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Integral Method ◽  
Special Functions ◽  
Incomplete Gamma Function ◽  
Hurwitz Zeta Function ◽  
Contour Integral ◽  
Fundamental Constants ◽  
Infinite Sum

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

scholarly journals Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

scholarly journals A Note on the Summation of the Incomplete Gamma Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2369
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Zeta Functions ◽  
Incomplete Gamma Function ◽  
Zero Distribution ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Almost All ◽  
Infinite Sum

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

scholarly journals Error bounds for the asymptotic expansion of the Hurwitz zeta function

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170363 ◽  
Author(s):  
G. Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Zeta Function ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Remainder Term ◽  
Hurwitz Zeta Function ◽  
Polygamma Functions

In this paper, we reconsider the large- a asymptotic expansion of the Hurwitz zeta function ζ ( s , a ). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G -function and the s -derivative of the Hurwitz zeta function ζ ( s , a ) are provided. A detailed discussion on the sharpness of our error bounds is also given.

scholarly journals A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 9
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Parabolic Cylinder ◽  
Fundamental Constants ◽  
Lerch Zeta Function ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral ◽  
Almost All

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

scholarly journals A procedure for generating infinite series identities

2004 ◽  
Vol 2004 (67) ◽  
pp. 3653-3662
Author(s):  
Anthony A. Ruffa
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Infinite Series ◽  
Special Functions ◽  
Hurwitz Zeta Function ◽  
Gauss Hypergeometric Function ◽  
Polygamma Function ◽  
Polygamma Functions ◽  
Infinite Sums ◽  
Lerch Transcendent

A procedure for generating infinite series identities makes use of the generalized method of exhaustion by analytically evaluating the inner series of the resulting double summation. Identities are generated involving both elementary and special functions. Infinite sums of special functions include those of the gamma and polygamma functions, the Hurwitz Zeta function, the polygamma function, the Gauss hypergeometric function, and the Lerch transcendent. The procedure can be automated withMathematica(or equivalent software).

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