Hadamard's gamma function

Extension of the factorial function
Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles.

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 than Euler's gamma function. It is defined as:

H ( x ) = 1 Γ ( 1 − x ) d d x { ln ⁡ ( Γ ( 1 2 − x 2 ) Γ ( 1 − x 2 ) ) } , {\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\},}

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

H ( n ) = Γ ( n ) = ( n − 1 ) ! {\displaystyle H(n)=\Gamma (n)=(n-1)!}

Properties

Unlike the classical gamma function, Hadamard's gamma function H(x) is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation

H ( x + 1 ) = x H ( x ) + 1 Γ ( 1 − x ) , {\displaystyle H(x+1)=xH(x)+{\frac {1}{\Gamma (1-x)}},}

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

Representations

Hadamard's gamma can also be expressed as

H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) = Φ ( − 1 , 1 , − x ) Γ ( − x ) {\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}={\frac {\Phi \left(-1,1,-x\right)}{\Gamma (-x)}}}

where Φ {\displaystyle \Phi } is the Lerch zeta function, and as

H ( x ) = Γ ( x ) [ 1 + sin ⁡ ( π x ) 2 π { ψ ( x 2 ) − ψ ( x + 1 2 ) } ] , {\displaystyle H(x)=\Gamma (x)\left[1+{\frac {\sin(\pi x)}{2\pi }}\left\{\psi \left({\dfrac {x}{2}}\right)-\psi \left({\dfrac {x+1}{2}}\right)\right\}\right],}

where ψ(x) denotes the digamma function.

See also

  • Gamma function
  • Pseudogamma function

References

  • 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.