5 Gamma Function Properties 5.14 Multidimensional Integrals 5.16 Sums

§5.15 Polygamma Functions

Defines:: $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$ : polygamma functions
Keywords:: continued fractions, polygamma functions
Referenced by:: §5.4(ii)
Permalink:: http://dlmf.nist.gov/5.15

The functions $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$ , $n=1,2,\dots$ , are called the polygamma functions. In particular, ${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(z\right)$ is the trigamma function; ${\mathop{\psi\/}\nolimits^{{\prime\prime}}}$ , $\mathop{\psi^{{(3)}}\/}\nolimits$ , $\mathop{\psi^{{(4)}}\/}\nolimits$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. This includes asymptotic expansions: compare §§2.1(ii)–2.1(iii).

In (5.15.2)–(5.15.7) $n,m=1,2,3,\dots$ , and for $\mathop{\zeta\/}\nolimits\!\left(n+1\right)$ see §25.6(i).

5.15.1 ${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(z\right)=\sum_{{k=0}}^{\infty}% \frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$ ,

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/5.15.E1
Encodings:: TeX, pMML, png

5.15.2 $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(1\right)=(-1)^{{n+1}}n!\mathop{\zeta\/% }\nolimits\!\left(n+1\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, : : factorial, $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$ : polygamma functions and : nonnegative integer
A&S Ref:: 6.4.2
Referenced by:: §5.15
Permalink:: http://dlmf.nist.gov/5.15.E2
Encodings:: TeX, pMML, png

5.15.3 $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(\tfrac{1}{2}\right)=(-1)^{{n+1}}n!(2^{% {n+1}}-1)\mathop{\zeta\/}\nolimits\!\left(n+1\right),$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, : : factorial, $\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$ : polygamma functions and : nonnegative integer
A&S Ref:: 6.4.4
Permalink:: http://dlmf.nist.gov/5.15.E3
Encodings:: TeX, pMML, png

5.15.4 ${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2% }\pi^{2}-4\sum_{{k=1}}^{{n-1}}\frac{1}{(2k-1)^{2}},$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer and : nonnegative integer
A&S Ref:: 6.4.5
Permalink:: http://dlmf.nist.gov/5.15.E4
Encodings:: TeX, pMML, png

5.15.5 ${\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(z+1\right)={\mathop{\psi\/}\nolimits% ^{{(n)}}}\!\left(z\right)+(-1)^{{n}}n!z^{{-n-1}},$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, : nonnegative integer and : complex variable
A&S Ref:: 6.4.6
Permalink:: http://dlmf.nist.gov/5.15.E5
Encodings:: TeX, pMML, png

5.15.6 ${\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(1-z\right)+(-1)^{{n-1}}{\mathop{\psi% \/}\nolimits^{{(n)}}}\!\left(z\right)=(-1)^{{n}}\pi\frac{{d}^{n}}{{dz}^{n}}% \mathop{\cot\/}\nolimits\!\left(\pi z\right),$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer and : complex variable
A&S Ref:: 6.4.7
Permalink:: http://dlmf.nist.gov/5.15.E6
Encodings:: TeX, pMML, png

5.15.7 ${\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(mz\right)=\frac{1}{m^{{n+1}}}\sum_{{% k=0}}^{{m-1}}{\mathop{\psi\/}\nolimits^{{(n)}}}\!\left(z+\frac{k}{m}\right).$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer, : nonnegative integer, : nonnegative integer and : complex variable
A&S Ref:: 6.4.8
Referenced by:: §5.15
Permalink:: http://dlmf.nist.gov/5.15.E7
Encodings:: TeX, pMML, png

As $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta\;(<\pi)$

5.15.8 ${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(z\right)\sim\frac{1}{z}+\frac{1}{% 2z^{2}}+\sum_{{k=1}}^{\infty}\frac{\mathop{B_{{2k}}\/}\nolimits}{z^{{2k+1}}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\sim$ : asymptotic equality, : nonnegative integer and : complex variable
A&S Ref:: 6.4.12
Permalink:: http://dlmf.nist.gov/5.15.E8
Encodings:: TeX, pMML, png

For $\mathop{B_{{2k}}\/}\nolimits$ see §24.2(i).

For continued fractions for ${\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(z\right)$ and ${\mathop{\psi\/}\nolimits^{{\prime\prime}}}\!\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).

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