Hadamard's gamma function

In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function). This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way from Euler's gamma function. It is defined as:

H (x) = \frac{1}{Γ (1 - x)} \frac{d}{d x} {\ln (\frac{Γ (\frac{1}{2} - \frac{x}{2})}{Γ (1 - \frac{x}{2})})},

where $Γ(x)$ denotes the classical gamma function. If $n$ is a positive integer, then:

H (n) = Γ (n) = (n - 1)!

Properties

Unlike the classical gamma function, Hadamard's gamma function $H (x)$ is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation

H (x + 1) = x H (x) + \frac{1}{Γ (1 - x)},

with the understanding that $\frac{1}{Γ (1 - x)}$ is taken to be $0$ for positive integer values of $x$ .

The Hadamard's gamma function has a superadditive property:

H (x) + H (y) \leq H (x + y),

for all $x, y \geq α$ , where $α = 1.5031...$ (sequence A381340 in the OEIS) is the unique solution to the equation $H (2 t) = 2 H (t)$ in the interval $[1.5, \infty)$ .

Representations

Hadamard's gamma can also be expressed as

H (x) = \frac{ψ (1 - \frac{x}{2}) - ψ (\frac{1}{2} - \frac{x}{2})}{2 Γ (1 - x)} = \frac{L (- 1, 1, - x)}{Γ (- x)},

and also as

H (x) = Γ (x) [1 + \frac{\sin (π x)}{2 π} {ψ (\frac{x}{2}) - ψ (\frac{x + 1}{2})}],

where $ψ (x)$ denotes the digamma function, and $L$ denotes the Lerch zeta function.

References

↑ Alzer, Horst (January 8, 2009). "A superadditive property of Hadamard's gamma function". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 79: 11–23. doi:10.1007/s12188-008-0009-5. Retrieved November 18, 2025.

Hadamard, M. J. (1894), Sur L'Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière (PDF) (in French), Œuvres de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968
Srivastava, H. M.; Junesang, Choi (2012). Zeta and Q-Zeta Functions and Associated Series and Integrals. Elsevier insights. p. 124. ISBN 978-0-12-385218-2.
"Introduction to the Gamma Function". The Wolfram Functions Site. Wolfram Research, Inc. Retrieved 27 February 2016.

[1] Alzer, Horst (January 8, 2009). "A superadditive property of Hadamard's gamma function". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 79: 11–23. doi:10.1007/s12188-008-0009-5. Retrieved November 18, 2025.

Hadamard's gamma function

Properties

Representations

See also

References