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Trigamma function

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In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by

ψ ψ 1 ( z ) = d 2 d z 2 ln ⁡ ⁡ Γ Γ ( z ) {\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)} .

It follows from this definition that

ψ ψ 1 ( z ) = d d z ψ ψ ( z ) {\displaystyle \psi _{1}(z)={\frac {d}{dz}}\psi (z)}

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

ψ ψ 1 ( z ) = ∑ ∑ n = 0 ∞ ∞ 1 ( z + n ) 2 , {\displaystyle \psi _{1}(z)=\sum _{n=0}^{\infty }{\frac {1}{(z+n)^{2}}},}

making it a special case of the Hurwitz zeta function

ψ ψ 1 ( z ) = ζ ζ ( 2 , z ) . {\displaystyle \psi _{1}(z)=\zeta (2,z).}

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

Contents

• Calculation

• Recurrence and reflection formulae

• Special values

• Relation to the Clausen function

• Appearance

• See also

• Notes

• References

Calculation

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

ψ ψ 1 ( z ) = ∫ ∫ 0 1 ∫ ∫ 0 x x z − − 1 y ( 1 − − x ) d y d x {\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dy\,dx}

using the formula for the sum of a geometric series. Integration over y yields:

ψ ψ 1 ( z ) = − − ∫ ∫ 0 1 x z − − 1 ln ⁡ ⁡ x 1 − − x d x {\displaystyle \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx}

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

ψ ψ 1 ( z ) ∼ ∼ d d z ( ln ⁡ ⁡ z − − ∑ ∑ n = 1 ∞ ∞ B n n z n ) = 1 z + ∑ ∑ n = 1 ∞ ∞ B n z n + 1 = ∑ ∑ n = 0 ∞ ∞ B n z n + 1 = 1 z + 1 2 z 2 + 1 6 z 3 − − 1 30 z 5 + 1 42 z 7 − − 1 30 z 9 + 5 66 z 11 − − 691 2730 z 13 + 7 6 z 15 ⋯ ⋯ {\displaystyle {\begin{aligned}\psi _{1}(z)&\sim {\operatorname {d} \over \operatorname {d} \!z}\left(\ln z-\sum _{n=1}^{\infty }{\frac {B_{n}}{nz^{n}}}\right)\\&={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {B_{n}}{z^{n+1}}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{z^{n+1}}}\\&={\frac {1}{z}}+{\frac {1}{2z^{2}}}+{\frac {1}{6z^{3}}}-{\frac {1}{30z^{5}}}+{\frac {1}{42z^{7}}}-{\frac {1}{30z^{9}}}+{\frac {5}{66z^{11}}}-{\frac {691}{2730z^{13}}}+{\frac {7}{6z^{15}}}\cdots \end{aligned}}}

where Bn is the nth Bernoulli number and we choose B1 = ⁠1/2⁠.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

ψ ψ 1 ( z + 1 ) = ψ ψ 1 ( z ) − − 1 z 2 {\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}}

and the reflection formula

ψ ψ 1 ( 1 − − z ) + ψ ψ 1 ( z ) = π π 2 sin 2 ⁡ ⁡ π π z {\displaystyle \psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}\pi z}}\,}

which immediately gives the value for z = ⁠1/2⁠: ψ ψ 1 ( 1 2 ) = π π 2 2 {\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}} .

Special values

At positive integer values we have that

ψ ψ 1 ( n ) = π π 2 6 − − ∑ ∑ k = 1 n − − 1 1 k 2 , ψ ψ 1 ( 1 ) = π π 2 6 , ψ ψ 1 ( 2 ) = π π 2 6 − − 1 , ψ ψ 1 ( 3 ) = π π 2 6 − − 5 4 . {\displaystyle \psi _{1}(n)={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}},\qquad \psi _{1}(1)={\frac {\pi ^{2}}{6}},\qquad \psi _{1}(2)={\frac {\pi ^{2}}{6}}-1,\qquad \psi _{1}(3)={\frac {\pi ^{2}}{6}}-{\frac {5}{4}}.}

At positive half integer values we have that

ψ ψ 1 ( n + 1 2 ) = π π 2 2 − − 4 ∑ ∑ k = 1 n 1 ( 2 k − − 1 ) 2 , ψ ψ 1 ( 1 2 ) = π π 2 2 , ψ ψ 1 ( 3 2 ) = π π 2 2 − − 4. {\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}},\qquad \psi _{1}\left({\tfrac {1}{2}}\right)={\frac {\pi ^{2}}{2}},\qquad \psi _{1}\left({\tfrac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4.}

The trigamma function has other special values such as:

ψ ψ 1 ( 1 4 ) = π π 2 + 8 G {\displaystyle \psi _{1}\left({\tfrac {1}{4}}\right)=\pi ^{2}+8G}

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.

Relation to the Clausen function

The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,cite-ref-1[1]

ψ ψ 1 ( p q ) = π π 2 2 sin 2 ⁡ ⁡ ( π π p / q ) + 2 q ∑ ∑ m = 1 ( q − − 1 ) / 2 sin ⁡ ⁡ ( 2 π π m p q ) Cl 2 ( 2 π π m q ) . {\displaystyle \psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).}

Appearance

The trigamma function appears in this sum formula:cite-ref-mezo-2-0[2]

∑ ∑ n = 1 ∞ ∞ n 2 − − 1 2 ( n 2 + 1 2 ) 2 ( ψ ψ 1 ( n − − i 2 ) + ψ ψ 1 ( n + i 2 ) ) = − − 1 + 2 4 π π coth ⁡ ⁡ π π 2 − − 3 π π 2 4 sinh 2 ⁡ ⁡ π π 2 + π π 4 12 sinh 4 ⁡ ⁡ π π 2 ( 5 + cosh ⁡ ⁡ π π 2 ) . {\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left(\psi _{1}{\bigg (}n-{\frac {i}{\sqrt {2}}}{\bigg )}+\psi _{1}{\bigg (}n+{\frac {i}{\sqrt {2}}}{\bigg )}\right)=-1+{\frac {\sqrt {2}}{4}}\pi \coth {\frac {\pi }{\sqrt {2}}}-{\frac {3\pi ^{2}}{4\sinh ^{2}{\frac {\pi }{\sqrt {2}}}}}+{\frac {\pi ^{4}}{12\sinh ^{4}{\frac {\pi }{\sqrt {2}}}}}\left(5+\cosh \pi {\sqrt {2}}\right).}