Generalized Pochhammer symbol

In mathematics, the generalized Pochhammer symbol of parameter [math]\displaystyle{ \alpha\gt 0 }[/math] and partition [math]\displaystyle{ \kappa=(\kappa_1,\kappa_2,\ldots,\kappa_m) }[/math] generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as

[math]\displaystyle{ (a)^{(\alpha )}_\kappa=\prod_{i=1}^m \prod_{j=1}^{\kappa_i} \left(a-\frac{i-1}{\alpha}+j-1\right). }[/math]

It is used in multivariate analysis.

References

• Dunkl, Charles F.; Xu, Yuan (2001), Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, 81, Cambridge University Press, p. 308, ISBN 9780521800433, https://books.google.com/books?id=bFPIC7DAHdsC&pg=PA308