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Trigamma function

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Mathematical function For Barnes's gamma function of 3 variables, see triple gamma function.
Color representation of the trigamma function, ψ1(z), in a rectangular region of the complex plane. It is generated using the domain coloring method.

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

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

It follows from this definition that

ψ 1 ( z ) = d d z 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 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 1 ( z ) = 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.

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

and the reflection formula

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

Special values

At positive integer values we have that

ψ 1 ( n ) = π 2 6 − ∑ k = 1 n − 1 1 k 2 , ψ 1 ( 1 ) = π 2 6 , ψ 1 ( 2 ) = π 2 6 − 1 , ψ 1 ( 3 ) = 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 1 ( n + 1 2 ) = π 2 2 − 4 ∑ k = 1 n 1 ( 2 k − 1 ) 2 , ψ 1 ( 1 2 ) = π 2 2 , ψ 1 ( 3 2 ) = 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 1 ( 1 4 ) = 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,

ψ 1 ( p q ) = π 2 2 sin 2 ⁡ ( π p / q ) + 2 q ∑ m = 1 ( q − 1 ) / 2 sin ⁡ ( 2 π m p q ) Cl 2 ( 2 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:

∑ 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 ) . {\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).}

See also

Notes

  1. Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.
  2. Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation. 219 (18): 9838–9846. doi:10.1016/j.amc.2013.03.122.

References

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