Polygamma function

At all the nonpositive integers these polygamma functions have a pole of order m + 1.

This expresses the polygamma function as the Laplace transform of ⁠(−1)m+1 tm/1 − e−t⁠.

Setting m = 0 in the above formula does not give an integral representation of the digamma function.

The digamma function has an integral representation, due to Gauss, which is similar to the m = 0 case above but which has an extra term ⁠e−t/t⁠.

It satisfies the recurrence relation which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers: and for all

uniquely to positive real numbers only due to their recurrence relation and one given function-value, say ψ(m)(1), except in the case m = 0 where the additional condition of strict monotonicity on

This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on

The case m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).

They obey the recursion equation The multiplication theorem gives and for the digamma function.

This representation can be written more compactly in terms of the Hurwitz zeta function as This relation can for example be used to compute the special values[1] Alternately, the Hurwitz zeta can be understood to generalize the polygamma to arbitrary, non-integer order.

Also the Lerch transcendent can be denoted in terms of polygamma function The Taylor series at z = -1 is and which converges for |z| < 1.

These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:[2] and where we have chosen B1 = ⁠1/2⁠, i.e. the Bernoulli numbers of the second kind.

The hyperbolic cotangent satisfies the inequality and this implies that the function is non-negative for all m ≥ 1 and t ≥ 0.

The convexity inequality et ≥ 1 + t implies that is non-negative for all m ≥ 1 and t ≥ 0, so a similar Laplace transformation argument yields the complete monotonicity of Therefore, for all m ≥ 1 and x > 0, Since both bounds are strictly positive for

Graphs of the polygamma functions ψ , ψ (1) , ψ (2) and ψ (3) of real arguments
Plot of the digamma function, the first polygamma function, in the complex plane, with colors showing one cycle of phase shift around each pole and zero
Plot of the digamma function , the first polygamma function, in the complex plane from −2−2i to 2+2i with colors created by Mathematica's function ComplexPlot3D showing one cycle of phase shift around each pole and the zero