Polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers formula_1 defined as the th derivative of the logarithm of the gamma function: Thus holds where is the digamma function and is the gamma function. They are holomorphic on formula_4. At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function. Integral representation. When and , the polygamma function equals This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and  non-negative, is a completely monotone function. Setting 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 case above but which has an extra term . Recurrence relation. 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 . Like the log-gamma function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their  recurrence relation and one given function-value, say , except in the case where the additional condition of strict monotonicity on formula_9 is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on formula_9 is demanded additionally. The case must be treated differently because is not normalizable at infinity (the sum of the reciprocals doesn't converge). Reflection relation. where is alternately an odd or even polynomial of degree with integer coefficients and leading coefficient . They obey the recursion equation Multiplication theorem. The multiplication theorem gives and for the digamma  function. Series representation. The polygamma function has the series representation which holds for integer values of and any complex not equal to a negative integer. This representation can be written more compactly in terms of the Hurwitz zeta function as 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, This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as: Now, the natural logarithm of the gamma function is easily representable: Finally , we arrive at a summation representation for the polygamma function: Where is the Kronecker delta. Also the Lerch transcendent can be denoted in terms of polygamma function Taylor series. The Taylor series at is and which converges for . Here, 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: and where we  have chosen , i.e. the Bernoulli numbers of the second kind. Inequalities. The hyperbolic cotangent satisfies the inequality and this implies that the function is non-negative for all and . It follows that the Laplace transform of this function is completely monotone. By the integral representation above, we conclude that is completely monotone. The convexity inequality implies that is non-negative for all and , so a similar Laplace transformation argument yields the complete monotonicity of Therefore, for all and ,