Carlson symmetric form

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, the polygamma function of order m is a meromorphic function on $\mathbb {C}$ and defined as the (m+1)-th derivative of the logarithm of the gamma function:

$\psi ^{(m)}(z):={\frac {d^{m}}{dz^{m}}}\psi (z)={\frac {d^{m+1}}{dz^{m+1}}}\ln \Gamma (z).$

Thus

$\psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}$

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on $\mathbb {C} \setminus -\mathbb {N} _{0}$. At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ⁽¹⁾(z) is sometimes called the trigamma function.

The logarithm of the gamma function and the first few polygamma functions in the complex plane

$\ln \Gamma (z)$	$\psi ^{(0)}(z)$	$\psi ^{(1)}(z)$
$\psi ^{(2)}(z)$	$\psi ^{(3)}(z)$	$\psi ^{(4)}(z)$

Integral representation

The polygamma function may be represented as

$\psi ^{(m)}(z)=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-zt}}{1-e^{-t}}}dt$

which holds for Re z >0 and m > 0. For m = 0 see the digamma function definition.

Recurrence relation

It satisfies the recurrence relation

$\psi ^{(m)}(z+1)=\psi ^{(m)}(z)+{\frac {(-1)^{m}\,m!}{z^{m+1}}}$

which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:

${\frac {\psi ^{(m)}(n)}{(-1)^{m+1}\,m!}}=\zeta (1+m)-\sum _{k=1}^{n-1}{\frac {1}{k^{m+1}}}=\sum _{k=n}^{\infty }{\frac {1}{k^{m+1}}}\qquad m\geq 1$

and

$\psi ^{(0)}(n)=-\gamma \ +\sum _{k=1}^{n-1}{\frac {1}{k}}$

for all $n\in \mathbb {N}$. Like the $\ln \Gamma$-function, the polygamma functions can be generalized from the domain $\mathbb {N}$ uniquely to positive real numbers only due to their recurrence relation and one given function-value, say $\psi ^{(m)}(1)$, except in the case m=0 where the additional condition of strictly monotony on $\mathbb {R} ^{+}$ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on $\mathbb {R} ^{+}$ is demanded additionally. The case m=0 must be treated differently because $\psi ^{(0)}$ is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relation

$(-1)^{m}\psi ^{(m)}(1-z)-\psi ^{(m)}(z)=\pi {\frac {d^{m}}{dz^{m}}}\cot {(\pi z)}=\pi ^{m+1}{\frac {P_{m}(\cos(\pi z))}{\sin ^{m+1}(\pi z)}}$

where $P_{m}$ is alternatingly an odd resp. even polynomial of degree $|m-1|$ with integer coefficients and leading coefficient $(-1)^{m}\lceil 2^{m-1}\rceil$. They obey the recursion equation $P_{m+1}(x)=-\left((m+1)xP_{m}(x)+(1-x^{2})P_{m}^{\prime }(x)\right)$ with $P_{0}(x)=x$.

Multiplication theorem

The multiplication theorem gives

$k^{m+1}\psi ^{(m)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m)}\left(z+{\frac {n}{k}}\right)\qquad m\geq 1$

and

$k\psi ^{(0)}(kz)=k\log(k))+\sum _{n=0}^{k-1}\psi ^{(0)}\left(z+{\frac {n}{k}}\right)$

for the digamma function.

Series representation

The polygamma function has the series representation

$\psi ^{(m)}(z)=(-1)^{m+1}\;m!\;\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}$

which holds for m > 0 and any complex z not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as

$\psi ^{(m)}(z)=(-1)^{m+1}\;m!\;\zeta (m+1,z).$

Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

One more series may be permitted for the polygamma functions. As given by Schlömilch,

$1/\Gamma (z)=z\;{\mbox{e}}^{\gamma z}\;\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)\;{\mbox{e}}^{-z/n}$. This is a result of the Weierstrass factorization theorem.

Thus, the gamma function may now be defined as:

$\Gamma (z)={\frac {{\mbox{e}}^{-\gamma z}}{z}}\;\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}\;{\mbox{e}}^{z/n}$

Now, the natural logarithm of the gamma function is easily representable:

$\ln \Gamma (z)=-\gamma z-\ln(z)+\sum _{n=1}^{\infty }\left({\frac {z}{n}}-\ln(1+{\frac {z}{n}})\right)$

Finally, we arrive at a summation representation for the polygamma function:

$\psi ^{(n)}(z)={\frac {d^{n+1}}{dz^{n+1}}}\ln \Gamma (z)=-\gamma \delta _{n0}\;-\;{\frac {(-1)^{n}n!}{z^{n+1}}}\;+\;\sum _{k=1}^{\infty }\left({\frac {1}{k}}\delta _{n0}\;-\;{\frac {(-1)^{n}n!}{(k+z)^{n+1}}}\right)$

Where $\delta _{n0}$ is the Kronecker delta.

Taylor series

The Taylor series at z = 1 is

$\psi ^{(m)}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}{\frac {(m+k)!}{k!}}\;\zeta (m+k+1)\;z^{k}\qquad m\geq 1$

and

$\psi ^{(0)}(z+1)=-\gamma +\sum _{k=1}^{\infty }(-1)^{k+1}\zeta (k+1)\;z^{k}$

which converges for |z| < 1. Here, ζ is the Riemann zeta function. This series is easily derived from the corresponding Taylor series for the Hurwitz zeta function. This series may be used to derive a number of rational zeta series.

Asymptotic expansion

These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:

$\psi ^{(m)}(z)=(-1)^{m+1}\sum _{k=0}^{\infty }{\frac {(k+m-1)!}{k!}}{\frac {B_{k}}{z^{k+m}}}\qquad m\geq 1$

and

$\psi ^{(0)}(z)=\ln(z)-\sum _{k=1}^{\infty }{\frac {B_{k}}{kz^{k}}}$

where we have chosen $B_{1}=1/2$, i.e. the Bernoulli numbers of the second kind.

See also

• Factorial
• Gamma function
• Digamma function
• Trigamma function
• Generalized polygamma function

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 978-0-486-61272-0 . See section §6.4