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Hadamard's gamma function

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In mathematics, the Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from Gamma 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 to Euler's Gamma function. It is defined as:

Contents

H ( x ) = 1 Γ ( 1 − x ) d d x { ln ⁡ ( Γ ( 1 2 − x 2 ) Γ ( 1 − x 2 ) ) }

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

H ( 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 )

Representations

Hadamard's gamma can be expressed in terms of digamma functions as

H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x )

and as

H ( x ) = Γ ( x ) [ 1 + sin ⁡ ( π x ) 2 π { ψ ( x 2 ) − ψ ( x + 1 2 ) } ] ,

where ψ(x) denotes the digamma function.

References

Hadamard's gamma function Wikipedia
(Text) CC BY-SA


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