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This study introduces a novel bilateral generating function using Chebyshev polynomials and the incomplete gamma function. The research details methods for deriving these functions and explores special cases for broader applications.

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Area of Science:

  • Mathematics
  • Special Functions
  • Approximation Theory

Background:

  • Generating functions are crucial in various mathematical fields.
  • Chebyshev polynomials have wide applications in numerical analysis and approximation.
  • The incomplete gamma function is a key special function with diverse uses.

Purpose of the Study:

  • To derive a novel bilateral generating function.
  • To express this function using Chebyshev polynomials and the incomplete gamma function.
  • To summarize and derive generating functions for Chebyshev polynomials.

Main Methods:

  • Contour integral method for function derivation.
  • Double series representation for the generating function.
  • Evaluation of special cases involving Chebyshev polynomials and incomplete gamma function.

Main Results:

  • A new bilateral generating function is derived.
  • The function is presented as a double series involving Chebyshev polynomials and the incomplete gamma function.
  • Generating functions for Chebyshev polynomials are summarized and derived.

Conclusions:

  • The derived bilateral generating function offers a new tool in mathematical analysis.
  • The study provides a unified approach to generating functions involving Chebyshev polynomials and the incomplete gamma function.
  • Special case evaluations highlight the function's versatility.