Get the answer to "The Taylor Series of Gamma Function exists or..."

Barrett Osborn

Answered question

2022-11-22

The Taylor Series of Gamma Function exists or not.

x0xkatiex0xz6u

Beginner2022-11-23Added 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} = \frac{{[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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