The Taylor Series of Gamma Function exists or not.

Barrett Osborn

Barrett Osborn

Answered

2022-11-22

The Taylor Series of Gamma Function exists or not.

Answer & Explanation

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x0xkatiex0xz6u

Expert

2022-11-23Added 11 answers

If you want the Taylor series, you basically need the n t h derivative of Γ ( x ). These express in terms of the polygamma function. Considering
d n = [ Γ ( x ) ] ( n ) Γ ( x )
the first terms are
d 1 = ψ ( 0 ) ( x )
d 2 = ψ ( 0 ) ( x ) 2 + ψ ( 1 ) ( x )
d 3 = ψ ( 0 ) ( x ) 3 + 3 ψ ( 1 ) ( x ) ψ ( 0 ) ( x ) + ψ ( 2 ) ( x )
d 4 = ψ ( 0 ) ( x ) 4 + 6 ψ ( 1 ) ( x ) ψ ( 0 ) ( x ) 2 + 4 ψ ( 2 ) ( x ) ψ ( 0 ) ( x ) + 3 ψ ( 1 ) ( x ) 2 + ψ ( 3 ) ( x )
which "simplify" (a little !) when you perform the expansion around x = a, a being a positive integer.

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