combine/Psi - Maple Programming Help

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combine/Psi

combine Psi functions

Calling Sequence

 combine(f, Psi) $\mathrm{combine}\left(f,\mathrm{\Psi }\right)$

Parameters

 f - any expression

Description

 • Expressions involving Psi are combined as follows

$\mathrm{\Psi }\left(n±x\right)+\mathrm{\Psi }\left(m±x\right)⇒\mathrm{\Psi }\left(x\right)±Q\left(x\right)±\mathrm{\pi }P\left(\mathrm{cot}\left(\mathrm{\pi }x\right)\right)$

 where Q is a rational function, P a univariate polynomial and m and n are rationals.
 • This is done by applying the following recurrence and reflection formulae.

$\mathrm{\Psi }\left(1+x\right)=\mathrm{\Psi }\left(x\right)+\frac{1}{x}$

$\mathrm{\Psi }\left(1-x\right)=\mathrm{\Psi }\left(x\right)+\mathrm{\pi }\mathrm{cot}\left(\mathrm{\pi }x\right)$

$\mathrm{\Psi }\left(n,1+x\right)=\mathrm{\Psi }\left(n,x\right)+\frac{{\left(-1\right)}^{n}n!}{{x}^{n+1}}$

$\mathrm{\Psi }\left(n,1-x\right)={\left(-1\right)}^{n}\left(\mathrm{\Psi }\left(n,x\right)+\frac{{ⅆ}^{n}}{ⅆ{x}^{n}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\mathrm{\pi }\mathrm{cot}\left(\mathrm{\pi }x\right)\right)\right)$

 • You can enter the command combine/Psi using either the 1-D or 2-D calling sequence. For example, combine(Psi(x) + Psi(1+x), Psi) is equivalent to $\mathrm{combine}\left(\mathrm{\Psi }\left(x\right)+\mathrm{\Psi }\left(1+x\right),\mathrm{\Psi }\right)$.

Examples

 > $\mathrm{combine}\left(\mathrm{Ψ}\left(x\right)+\mathrm{Ψ}\left(1+x\right),\mathrm{Ψ}\right)$
 ${2}{}{\mathrm{Ψ}}{}\left({x}\right){+}\frac{{1}}{{x}}$ (1)
 > $\mathrm{combine}\left(\mathrm{Ψ}\left(1-x\right)-\mathrm{Ψ}\left(x\right)+\mathrm{Ψ}\left(y\right),\mathrm{Ψ}\right)$
 ${\mathrm{π}}{}{\mathrm{cot}}{}\left({\mathrm{π}}{}{x}\right){+}{\mathrm{Ψ}}{}\left({y}\right)$ (2)
 > $\mathrm{combine}\left(\mathrm{Ψ}\left(2,x-\frac{1}{2}\right)-\mathrm{Ψ}\left(2,x+\frac{1}{2}\right),\mathrm{Ψ}\right)$
 ${-}\frac{{2}}{{\left({x}{-}\frac{{1}}{{2}}\right)}^{{3}}}$ (3)
 > $\mathrm{combine}\left(\mathrm{Ψ}\left(1,x-\frac{1}{2}\right)-\mathrm{Ψ}\left(1,x+\frac{1}{2}\right),\mathrm{Ψ}\right)$
 $\frac{{1}}{{\left({x}{-}\frac{{1}}{{2}}\right)}^{{2}}}$ (4)