DLMF: GA.19 Mathematical Applications

§GA.19. Mathematical Applications

§GA.19(i)Summation of Rational Functions
§GA.19(ii)Mellin-Barnes Integrals
§GA.19(iii) $n$ -Dimensional Sphere

§GA.19(i). Summation of Rational Functions

Keywords:: gamma functions

As shown in Temme(1996)(§3.4), the results given in §GA.7(ii) can be used to sum infinite series of rational functions.

Example

GA.19.1 $S=\sum _{{k=0}}^{\infty}a_{k},$ $a_{k}=\frac{k}{(3k+2)(2k+1)(k+1)}.$

Notations:: $k$ : nonnegative integer and $a$ : real or complex variable
Encodings:: pMathML, pMathML, png, png, TeX, TeX

By decomposition into partial fractions (§)

GA.19.2 $a_{k}=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(\frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+\frac{2}{3}}\right).$

Notations:: $k$ : nonnegative integer and $a$ : real or complex variable
Encodings:: pMathML, png, TeX

Hence from (GA.7.6), (GA.4.13), and (GA.4.19)

GA.19.3 $S=\psi\!\left(\tfrac{1}{2}\right)-2\psi\!\left(\tfrac{2}{3}\right)-\EulerConstant=3\mathrm{ln}3-2\mathrm{ln}2-\tfrac{1}{3}\pi\sqrt{3}.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $\EulerConstant$ : Euler's constant
Encodings:: pMathML, png, TeX

§GA.19(ii). Mellin-Barnes Integrals

Keywords:: gamma function, Mellin-Barnes integral representation

Many special functions $f(z)$ can be represented as a Mellin-Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$ , the integration contour being doubly-infinite and eventually parallel to the imaginary axis. The left-hand side of (GA.13.1) is a typical example. By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$ , or small $|z|$ , can be obtained complete with an integral representation of the error term. For further information and examples see § and Paris and Kaminski(2001)(Chapters 5, 6, and 8).