TOPICS
Search

q-Polygamma Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The q-digamma function psi_q(z), also denoted psi_q^((0))(z), is defined as

psi_q(z)=1/(Gamma_q(z))(partialGamma_q(z))/(partialz),
(1)

where Gamma_q(z) is the q-gamma function. It is also given by the sum

psi_q(z)=-ln(1-q)+lnqsum_(n=0)^infty(q^(n+z))/(1-q^(n+z)).
(2)

The q-polygamma function psi_q^n(z) (also denoted psi_q^((n))(z)) is defined by

psi_q^((n))(z)=(partial^npsi_q(z))/(partialz^n).
(3)

It is implemented in the Wolfram Language as QPolyGamma[n, z, q], with the q-digamma function implemented as the special case QPolyGamma[z, q].

Certain classes of sums can be expressed in closed form using the q-polygamma function, including

sum_(k=1)^(infty)1/(1-a^k)=(psi_(1/a)(1)+ln(a-1)+ln(1/a))/(lna)
(4)
sum_(k=0)^(infty)1/(coshk+1)=2[1-psi_e^((1))(-ipi)].
(5)

The q-polygamma functions are related to the Lambert series

L(beta)=sum_(n=1)^(infty)(beta^n)/(1-beta^n)
(6)
=sum_(n=1)^(infty)1/(beta^(-n)-1)
(7)
=(psi_q(1)+ln(1-q))/(lnq)
(8)

(Borwein and Borwein 1987, pp. 91 and 95).

An identity connecting q-polygamma to elliptic functions is given by

pi-i[psi_(phi^2)^((0))(1/2-(ipi)/(4lnphi))-psi_(phi^2)^((0))(1/2+(ipi)/(4lnphi))] =-(lnphi)theta_2^2(phi^(-2)),
(9)

where phi is the golden ratio and theta_n(q) is an Jacobi theta function.

See also

False Logarithmic Series, Polygamma Function

Explore with Wolfram|Alpha

References

Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 1987.

Referenced on Wolfram|Alpha

q-Polygamma Function

Cite this as:

Weisstein, Eric W. "q-Polygamma Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-PolygammaFunction.html

Subject classifications