Pochhammer Symbol


The Pochhammer symbol


(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for n>=0 is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].

In combinatorics, the notation x^((n)) (Roman 1984, p. 5), <x>_n (Comtet 1974, p. 6), or x^(n^_) (Graham et al. 1994, p. 48) is used for the rising factorial, while (x)_n or x^(n__) denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations (x)_n and x^((n)).

The first few values of (x)_n for nonnegative integers n are


(OEIS A054654).

In closed form, (x)_n can be written


where s(n,k) is a Stirling number of the first kind.

The Pochhammer symbol satisfies


the dimidiation formulas


and the duplications formula

 (2x)_n={2^n(x)_(n/2)(x+1/2)_(n/2)   for n even; 2^n(x)_((n+1)/2)(x+1/2)_((n-1)/2)   for n odd

(Boros and Moll 2004, p. 17).

A ratio of Pochhammer symbols is given in closed form by

 ((x)_n)/((x)_m)={(x+m)_(n-m)   if n>=m; 1/((x+n)_(m-n))   if n<=m

(Boros and Moll 2004, p. 17).

The derivative is given by


where psi_0(x) is the digamma function.

Special values include


The Pochhammer symbol (x)_n obeys the transformation due to Euler


where Delta is the forward difference and

 Delta^ka_0=sum_(m=0)^k(-1)^m(k; m)a_(k-m)

(Nørlund 1955).

The sum of 1/(k)_p can be done in closed form as


for p>1.


Consider the product


This function converges to 0, to a finite value, or diverges, depending on the value of z. The critical curve is given by the implicit equation


Inside this curve, the function converges to 0, whereas outside it, it diverges. The maximum real value at which convergence occurs is given by x_+=0.54221... (OEIS A090462), and the minimum value by x_-=-(1+x_+). The extremal values of y are given by y_+/-=+/-0.95883... (OEIS A090463). On the critical contour, f(z) takes on the value


Plotting a suitably scaled version of f(z) with k finite shows beautiful and subtle structures such as those illustrated above for k=100 (M. Trott, pers. comm., Dec. 1, 2003).


Another beautiful visualization plots sin(arg(f(z))), as illustrated above for k=2048 (M. Trott, pers. comm., Dec. 2, 2003).

See also

Factorial, Falling Factorial, Generalized Hypergeometric Function, Hankel's Symbol, Harmonic Logarithm, Hypergeometric Function, Kramp's Symbol, Rising Factorial

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, 1974.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 52, 1981.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Nørlund, N. E. "Hypergeometric Functions." Acta Math. 94, 289-349, 1955.Roman, S. The Umbral Calculus. New York: Academic Press, p. 5, 1984.Sloane, N. J. A. Sequences A054654, A090462, and A090463 in "The On-Line Encyclopedia of Integer Sequences."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.

Referenced on Wolfram|Alpha

Pochhammer Symbol

Cite this as:

Weisstein, Eric W. "Pochhammer Symbol." From MathWorld--A Wolfram Web Resource.

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