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This study derives new formulas for generalized double finite series using the Hurwitz-Lerch zeta function. These findings offer special cases involving trigonometric and gamma functions for mathematical research.

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Area of Science:

  • Mathematical Analysis
  • Special Functions Theory

Background:

  • The Hurwitz-Lerch zeta function is a significant special function with broad applications.
  • Generalized finite series require robust derivation methods for closed-form solutions.

Purpose of the Study:

  • To derive closed-form formulas for generalized double finite series.
  • To explore special cases involving the Hurwitz-Lerch zeta function, trigonometric functions, and the gamma function.

Main Methods:

  • Employing contour integration techniques.
  • Developing a generalized double finite series involving the Hurwitz-Lerch zeta function.

Main Results:

  • Established closed-form formulae in terms of special functions.
  • Identified specific summation and product formulae.
  • Generated a table of quotient gamma functions and accompanying plots.

Conclusions:

  • The contour integration method effectively derives generalized series formulas.
  • The results provide valuable special cases for the Hurwitz-Lerch zeta function and related functions.