Polygamma function

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In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:

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

Thus

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on . 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 Γ(z) ψ(0)(z) ψ(1)(z)
ψ(2)(z) ψ(3)(z) ψ(4)(z)

Integral representation

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When m > 0 and Re z > 0, the polygamma function equals

 

where   is the Hurwitz zeta function.

This expresses the polygamma function as the Laplace transform of (−1)m+1 tm/1 − et. It follows from Bernstein's theorem on monotone functions that, for m > 0 and x real and non-negative, (−1)m+1 ψ(m)(x) is a completely monotone function.

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 et/t.

Recurrence relation

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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  , where   is the Euler–Mascheroni constant. 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 ψ(m)(1), except in the case m = 0 where the additional condition of strict monotonicity on   is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on   is demanded additionally. The case m = 0 must be treated differently because ψ(0) is not normalizable at infinity (the sum of the reciprocals doesn't converge).

Reflection relation

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where Pm is alternately an odd or even polynomial of degree |m − 1| with integer coefficients and leading coefficient (−1)m⌈2m − 1. They obey the recursion equation

 

Multiplication theorem

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The multiplication theorem gives

 

and

 

for the digamma function.

Series representation

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The polygamma function has the series representation

 

which holds for integer values of 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

 

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.

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 δn0 is the Kronecker delta.

Also the Lerch transcendent

 

can be denoted in terms of polygamma function

 

Taylor series

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The Taylor series at z = -1 is

 

and

 

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

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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.

Inequalities

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The hyperbolic cotangent satisfies the inequality

 

and this implies that the function

 

is non-negative for all m ≥ 1 and t ≥ 0. 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 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  , we have:

  •   is strictly convex.
  • For  , the digamma function,  , is strictly monotonic increasing and strictly concave.
  • For   odd, the polygamma functions,  , are strictly positive, strictly monotonic decreasing and strictly convex.
  • For   even the polygamma functions,  , are strictly negative, strictly monotonic increasing and strictly concave.

This can be seen in the first plot above.

Trigamma bounds and asymptote

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For the case of the trigamma function ( ) the final inequality formula above for  , can be rewritten as:

 

so that for  :  .

See also

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References

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  1. ^ Kölbig, K. S. (1996). "The polygamma function psi^k(x) for x=1/4 and x=3/4". J. Comput. Appl. Math. 75 (1): 43–46. doi:10.1016/S0377-0427(96)00055-6.
  2. ^ Blümlein, J. (2009). "Structural relations of harmonic sums and Mellin transforms up to weight w=5". Comp. Phys. Comm. 180 (11): 2218–2249. arXiv:0901.3106. Bibcode:2009CoPhC.180.2218B. doi:10.1016/j.cpc.2009.07.004.