Pochhammer Symbol

A.k.a. Rising Factorial. For an Integer $n>0$,

$$(a)_n\equiv {\Gamma(a+k)\over\Gamma(a)} = a(a+1)\cdots(a+n-1),$$ (1)

where $\Gamma(z)$ is the Gamma Function and

$$(a)_0\equiv 1.$$ (2)

The Notation conflicts with both that for q-Series and that for Gaussian Coefficients, so context usually serves to distinguish the three. Additional identities are

$${d\over da} (a)_n = (a)_n[F(a+n-1)-F(a-1)]$$ (3)

$$(a)_{n+k} = (a+n)_k(a)_n,$$ (4)

where $F$ is the Digamma Function. The Pochhammer symbol arises in series expansions of Hypergeometric Functions and Generalized Hypergeometric Functions.

See also Factorial, Generalized Hypergeometric Function, Harmonic Logarithm, Hypergeometric Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 256, 1972.

Spanier, J. and Oldham, K. B. ``The Pochhammer Polynomials $(x)_n$.'' Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149-165, 1987.