Symbolic computations via Fourier–Legendre expansions and fractional operators

Abstract

We investigate the interplay between the Caputo fractional operators $D^{\pm 1/2}$ and Fourier–Legendre expansions of hypergeometric functions, resulting in transformation formulas for ${}_{q+1}{F}_q(z)$ series with quarter-integer parameters and in explicit evaluations of many series, including: $\begin{array}{ll}{\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{(-1{)}^n}{2n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{1}{4n+3},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^3\frac{1}{n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^3\frac{1}{(n+1{)}^2},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{H_{2n}}{2^n(n+1)},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{H_{2n}}{2^n(2n-1)}.}\end{array}$

Keywords:

• Hypergeometric function
• fractional calculus
• Fourier–Legendre expansion
• closed form

AMS Subject Classifications:

• 33C20
• 26A33
• 42C10

Acknowledgments

The second author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the author(s).