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: $\begin{array}{ll}{\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{(-1{)}^n}{2n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{1}{4n+3},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^3\frac{1}{n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^3\frac{1}{(n+1{)}^2},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{H_{2n}}{2^n(n+1)},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n!}\right)}^2\frac{H_{2n}}{2^n(2n-1)}.}\end{array}$

中文翻译：

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通过傅里叶-勒让德展开和小数运算符的符号计算

我们研究 Caputo 分数运算符之间的相互作用$D^{\pm 1/2}$和超几何函数的傅里叶-勒让德展开，得到${}_{q+1}{F}_q(z)$ 具有四分之一整数参数的系列和许多系列的显式评估，包括： $\begin{array}{ll}{\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^2\frac{(-1{)}^n}{2n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^2\frac{1}{4n+3},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^3\frac{1}{n+1},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^3\frac{1}{(n+1{)}^2},}\\ {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^2\frac{H_{2n}}{2^n(n+1)},}& {\displaystyle \sum _{n\ge 0}{\left(\frac{{\left(\frac{1}{2}\right)}_n}{n！}\right)}^2\frac{H_{2n}}{2^n(2n-1)}.}\end{array}$