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Trigamma function

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Trigamma function

In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by

Contents

ψ 1 ( z ) = d 2 d z 2 ln ⁡ Γ ( z ) .

It follows from this definition that

ψ 1 ( z ) = d d z ψ ( z )

where ψ(z) is the digamma function. It may also be defined as the sum of the series

ψ 1 ( z ) = ∑ n = 0 ∞ 1 ( z + n ) 2 ,

making it a special case of the Hurwitz zeta function

ψ 1 ( z ) = ζ ( 2 , z ) .

Note that the last two formulae are valid when 1 − z is not a natural number.

Calculation

A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

ψ 1 ( z ) = ∫ 0 1 ∫ 0 y x z − 1 y 1 − x d x d y

using the formula for the sum of a geometric series. Integration by parts yields:

ψ 1 ( z ) = − ∫ 0 1 x z − 1 ln ⁡ x 1 − x d x

An asymptotic expansion as a Laurent series is

ψ 1 ( z ) = 1 z + 1 2 z 2 + ∑ k = 1 ∞ B 2 k z 2 k + 1 = ∑ k = 0 ∞ B k z k + 1

if we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

ψ 1 ( z + 1 ) = ψ 1 ( z ) − 1 z 2

and the reflection formula

ψ 1 ( 1 − z ) + ψ 1 ( z ) = π 2 sin 2 ⁡ π z

which immediately gives the value for 1/z = 1/2.

Special values

The trigamma function has the following special values:

ψ 1 ( 1 4 ) = π 2 + 8 G ψ 1 ( 1 2 ) = π 2 2 ψ 1 ( 1 ) = π 2 6 ψ 1 ( 3 2 ) = π 2 2 − 4 ψ 1 ( 2 ) = π 2 6 − 1

where G represents Catalan's constant.

There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z < 0. Each such pair of roots approaches Re zn = −n + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.

Appearance

The trigamma function appears in this surprising sum formula:

∑ n = 1 ∞ n 2 − 1 2 ( n 2 + 1 2 ) 2 ( ψ 1 ( n − i 2 ) + ψ 1 ( n + i 2 ) ) = − 1 + 2 4 π coth ⁡ π 2 − 3 π 2 4 sinh 2 ⁡ π 2 + π 4 12 sinh 4 ⁡ π 2 ( 5 + cosh ⁡ π 2 ) .

References

Trigamma function Wikipedia
(Text) CC BY-SA