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Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions
Feng Qi1,2,3, Chuan-Jun Huang4
1Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010 Henan China.
Revista De La Real Academia De Ciencias Exactas, Fisicas Y Naturales. Serie A, Matematicas
|September 1, 2020
Summary
Researchers derived a general formula for higher-order derivatives of function ratios using binomial inversion. This formula computes combinatorial sums involving special functions like the beta and Gauss hypergeometric functions.
Area of Science:
- Mathematics
- Combinatorics
- Special Functions
Background:
- Combinatorial identities often involve complex sums.
- General formulas for derivatives of function ratios are valuable in mathematical analysis.
Purpose of the Study:
- To derive a general formula for higher-order derivatives of a ratio of two differentiable functions.
- To compute specific combinatorial sums using this new formula and established mathematical tools.
Main Methods:
- Application of the binomial inversion formula.
- Calculus techniques for higher-order derivatives.
- Evaluation of sums involving the beta function, polygamma functions, and the Gauss hypergeometric function.
Main Results:
- A general formula for higher-order derivatives of function ratios was established.
- Several combinatorial sums were computed in terms of the beta function and its partial derivatives, polygamma functions, the Gauss hypergeometric function, and a determinant.
- The derived results extend existing known results in combinatorics.
Conclusions:
- The study successfully generalized known combinatorial identities.
- The developed formula provides a powerful tool for analyzing and computing complex mathematical sums.
- The findings contribute to the field of combinatorics and mathematical analysis.
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