pochhammer - Maple Programming Help

Home : Support : Online Help : Mathematics : Special Functions : pochhammer

pochhammer

general pochhammer function

 Calling Sequence pochhammer(z, a)

Parameters

 z - expression a - expression

Description

 • The pochhammer symbol is defined for a positive integer n and complex number z as

$\mathrm{pochhammer}\left(z,n\right)=z\left(z+1\right)...\left(z+n-1\right)$

 This is extended analytically to complex $n$ by using the formula

$\mathrm{pochhammer}\left(z,a\right)=\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z\right)}$

 • At all points $\left(z,a\right)$ such that $z$ and $z+a$ are positive integers, this is equivalent to:

$\mathrm{pochhammer}\left(z,a\right)=\underset{t\to 0}{lim}\frac{\mathrm{\Gamma }\left(z+a+t\right)}{\mathrm{\Gamma }\left(z+t\right)}$

 In the case that $z$ is a non-positive integer, pochhammer(z,a) is defined by this limit.
 In the case that both $z$ and $z+a$ are non-positive integers, Maple also signals the invalid_operation numeric event, allowing the user to control this singular behavior by catching the event. See numeric_events for more information.

Examples

 > $\mathrm{pochhammer}\left(5,3\right)$
 ${210}$ (1)
 > $\mathrm{pochhammer}\left(z,2\right)$
 ${\mathrm{pochhammer}}{}\left({z}{,}{2}\right)$ (2)
 > $\mathrm{pochhammer}\left(z,-3\right)$
 $\frac{{1}}{{\mathrm{pochhammer}}{}\left({-}{3}{+}{z}{,}{3}\right)}$ (3)
 > $\mathrm{pochhammer}\left(2,I\right)$
 ${\mathrm{\Gamma }}{}\left({2}{+}{I}\right)$ (4)
 > $\mathrm{NumericStatus}\left(\mathrm{invalid_operation}=\mathrm{false}\right):$
 > $\mathrm{pochhammer}\left(-3,2\right)$
 ${6}$ (5)
 > $\mathrm{NumericStatus}\left(\mathrm{invalid_operation}=\mathrm{false}\right)$
 ${\mathrm{invalid_operation}}{=}{\mathrm{true}}$ (6)
 > $\mathrm{pochhammer}\left(0,0\right)$
 ${1}$ (7)
 > $\mathrm{NumericStatus}\left(\mathrm{invalid_operation}\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{diff}\left(\mathrm{pochhammer}\left(a,x\right),x\right)$
 ${\mathrm{pochhammer}}{}\left({a}{,}{x}\right){}{\mathrm{\Psi }}{}\left({x}{+}{a}\right)$ (9)
 > $\mathrm{diff}\left(\mathrm{pochhammer}\left(a,x\right),a\right)$
 ${\mathrm{pochhammer}}{}\left({a}{,}{x}\right){}\left({\mathrm{\Psi }}{}\left({x}{+}{a}\right){-}{\mathrm{\Psi }}{}\left({a}\right)\right)$ (10)
 > $\mathrm{series}\left(\mathrm{pochhammer}\left(a,x\right),x,3\right)$
 ${1}{+}{\mathrm{\Psi }}{}\left({a}\right){}{x}{+}\left(\frac{{\mathrm{\Psi }}{}\left({1}{,}{a}\right)}{{2}}{+}\frac{{{\mathrm{\Psi }}{}\left({a}\right)}^{{2}}}{{2}}\right){}{{x}}^{{2}}{+}{O}{}\left({{x}}^{{3}}\right)$ (11)
 > $\mathrm{pochhammer}\left(x,5\right)$
 ${\mathrm{pochhammer}}{}\left({x}{,}{5}\right)$ (12)
 > $\mathrm{expand}\left(\right)$
 ${{x}}^{{5}}{+}{10}{}{{x}}^{{4}}{+}{35}{}{{x}}^{{3}}{+}{50}{}{{x}}^{{2}}{+}{24}{}{x}$ (13)
 > $\mathrm{pochhammer}\left(2,\frac{1}{3}\right)$
 $\frac{{8}{}{\mathrm{\pi }}{}\sqrt{{3}}}{{27}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}$ (14)
 > $\mathrm{evalf}\left(\right)$
 ${1.190639350}$ (15)
 > $\mathrm{pochhammer}\left(-3.7+2.2I,1.5+2.7I\right)$
 ${-0.0005620896042}{+}{0.01961129135}{}{I}$ (16)
 > $\mathrm{convert}\left(\mathrm{pochhammer}\left(a,x\right),\mathrm{\Gamma }\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({x}{+}{a}\right)}{{\mathrm{\Gamma }}{}\left({a}\right)}$ (17)
 > $\mathrm{convert}\left(\mathrm{pochhammer}\left(a,x\right),\mathrm{binomial}\right)$
 $\left(\genfrac{}{}{0}{}{{a}{+}{x}{-}{1}}{{a}{-}{1}}\right){}{x}{!}$ (18)
 > $\mathrm{convert}\left(\mathrm{pochhammer}\left(a,x\right),\mathrm{factorial}\right)$
 $\frac{\left({a}{+}{x}{-}{1}\right){!}}{\left({a}{-}{1}\right){!}}$ (19)