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 ) = 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 is defined and analytic at all complex numbers. 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. The Hadamard's gamma function has a superadditive property:
H ( x ) + H ( y ) ≤ H ( x + y ) , {\displaystyle H(x)+H(y)\leq H(x+y),}
for all x , y ≥ α {\displaystyle x,y\geq \alpha } , where α = 1.5031... {\displaystyle \alpha =1.5031...} (sequence A381340 in the OEIS) is the unique solution to the equation H ( 2 t ) = 2 H ( t ) {\displaystyle H(2t)=2H(t)} in the interval [ 1.5 , ∞ ) {\displaystyle [1.5,\infty )} .
Representations Hadamard's gamma can also be expressed as
H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) = L ( − 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 {L\left(-1,1,-x\right)}{\Gamma (-x)}},}
and also as
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