Rising Factorial

The rising factorial , sometimes also denoted  (Comtet 1974, p. 6) or  (Graham et al. 1994, p. 48), is defined by

(1)

This function is also known as the rising factorial power (Graham et al. 1994, p. 48) and frequently called the Pochhammer symbol in the theory of special functions. The rising factorial is implemented in the Wolfram Language as Pochhammer[x, n].

The rising factorial is related to the gamma function  by

(2)

where

(3)

and is related to the falling factorial  by

(4)

The usual factorial is therefore related to the rising factorial by

(5)

for nonnegative integers  (Graham et al. 1994, p. 48).

Note that in combinatorial usage, the falling factorial is denoted  and the rising factorial is denoted  (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted  and the rising factorial is denoted  (Roman 1984, p. 5; Abramowitz and Stegun 1972, p. 256; Spanier 1987). Extreme caution is therefore needed in interpreting the meanings of the notations  and . In this work, the notation  is used for the rising factorial, despite the fact that Pochhammer symbol, which is another name for the rising factorial, is universally denoted .

The rising factorial arises in series expansions of hypergeometric functions and generalized hypergeometric functions.

The first few rising factorials are

			(6)
			(7)
			(8)
			(9)
			(10)

The derivative of the rising factorial is

(11)

where  is the digamma function.

REFERENCES:

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

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 101, 1999.

Roman, S. The Umbral Calculus. New York: Academic Press, p. 5, 1984.

Spanier, J. and Oldham, K. B. "The Pochhammer Polynomials ." Ch. 18 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 149-165, 1987.