pochhammer

R2026b

Pochhammer symbol

Syntax

pochhammer(x,n)

Description

pochhammer(x,n) returns the Pochhammer Symbol (x)_n.

Examples

Find Pochhammer Symbol for Numeric and Symbolic Inputs

Find the Pochhammer symbol for the numeric inputs x = 3 at n = 2.

pochhammer(3,2)

ans =
    12

Find the Pochhammer symbol for the symbolic input x at n = 3. The pochhammer function does not automatically return the expanded form of the expression. Use expand to force pochhammer to return the form of the expanded expression.

syms x
P = pochhammer(x, 3)
P = expand(P)

P =
pochhammer(x, 3)
P =
x^3 + 3*x^2 + 2*x

Rewrite and Factor Outputs of Pochhammer

If conditions are satisfied, expand rewrites the solution using gamma.

syms n x
assume(x>0)
assume(n>0)
P = pochhammer(x, n);
P = expand(P)

P =
gamma(n + x)/gamma(x)

To use the variables in further computations, clear their assumptions by recreating them using syms.

syms n x

To convert expanded output of pochhammer into its factors, use factor.

P = expand(pochhammer(x, 4));
P = factor(P)

P =
[ x, x + 3, x + 2, x + 1]

Differentiate Pochhammer Symbol

Differentiate pochhammer once with respect to x.

syms n x
diff(pochhammer(x,n),x)

ans =
pochhammer(x, n)*(psi(n + x) - psi(x))

Differentiate pochhammer twice with respect to n.

diff(pochhammer(x,n),n,2)

ans =
pochhammer(x, n)*psi(n + x)^2 + pochhammer(x, n)*psi(1, n + x)

Taylor Series Expansion of Pochhammer Symbol

Use taylor to find the Taylor series expansion of pochhammer with n = 3 around the expansion point x = 2.

syms x
taylor(pochhammer(x,3),x,2)

ans =
26*x + 9*(x - 2)^2 + (x - 2)^3 - 28

Plot Pochhammer Symbol

Plot the Pochhammer symbol from n = 0 to n = 4 for x. Use axis to display the region of interest.

syms x
fplot(pochhammer(x,0:4))
axis([-4 4 -4 4])

grid on
legend('n = 0','n = 1','n = 2','n = 3','n = 4','Location','Best')
title('Pochhammer symbol (x)_n for n=0 to n=4')

Input Arguments

x — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

n — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

More About

Pochhammer Symbol

Pochhammer’s symbol is defined as

$${\left(x\right)}_n=\frac{\Gamma \left(x+n\right)}{\Gamma \left(x\right)},$$

where Γ is the Gamma function.

If n is a positive integer, Pochhammer’s symbol is

$${\left(x\right)}_n=x\left(x+1\right)\mathrm{...}\left(x+n-1\right)$$

Algorithms

• If x and n are numerical values, then an explicit numerical result is returned. Otherwise, a symbolic function call is returned.

• If both x and x + n are nonpositive integers, then

  $${\left(x\right)}_n={\left(-1\right)}^n\frac{\Gamma \left(1-x\right)}{\Gamma \left(1-x-n\right)}.$$

• The following special cases are implemented.

  $$\begin{array}{l}{\left(x\right)}_0=1\\ {\left(x\right)}_1=x\\ {\left(x\right)}_{-1}=\frac{1}{x-1}\\ {\left(1\right)}_n=\Gamma \left(n+1\right)\\ {\left(2\right)}_n=\Gamma \left(n+2\right)\end{array}$$

• If n is a positive integer, then expand(pochhammer(x,n)) returns the expanded polynomial $$x\left(x+1\right)\mathrm{...}\left(x+n-1\right)$$.

• If n is not an integer, then expand(pochhammer(x,n)) returns a representation in terms of gamma.

Version History

Introduced in R2014b