Harman Patil (Editor)

Polygamma function

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit
Polygamma function

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

Contents

ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln ⁡ Γ ( z ) .

Thus

ψ ( 0 ) ( z ) = ψ ( z ) = Γ ′ ( z ) Γ ( z )

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on ℂ −ℕ0. 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.

Integral representation

The polygamma function may be represented as

ψ ( m ) ( z ) = ( − 1 ) m + 1 ∫ 0 ∞ t m e − z t 1 − e − t d t = − ∫ 0 1 t z − 1 1 − t ( ln ⁡ t ) m d t

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

Recurrence relation

It satisfies the recurrence relation

ψ ( m ) ( z + 1 ) = ψ ( m ) ( z ) + ( − 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:

ψ ( m ) ( n ) ( − 1 ) m + 1 m ! = ζ ( 1 + m ) − ∑ k = 1 n − 1 1 k m + 1 = ∑ k = n ∞ 1 k m + 1 m ≥ 1

and

ψ ( 0 ) ( n ) = − γ   + ∑ k = 1 n − 1 1 k

for all n ∈ ℕ. Like the log-gamma function, the polygamma functions can be generalized from the domain ℕ 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 monotony on ℝ+ is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on ℝ+ 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

( − 1 ) m ψ ( m ) ( 1 − z ) − ψ ( m ) ( z ) = π d m d z m cot ⁡ ( π z ) = π m + 1 P m ( cos ⁡ ( π z ) ) sin m + 1 ⁡ ( π z )

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

P 0 ( x ) = x P m + 1 ( x ) = − ( ( m + 1 ) x P m ( x ) + ( 1 − x 2 ) P m ′ ( x ) ) .

Multiplication theorem

The multiplication theorem gives

k m + 1 ψ ( m ) ( k z ) = ∑ n = 0 k − 1 ψ ( m ) ( z + n k ) m ≥ 1

and

k ψ ( 0 ) ( k z ) = k log ⁡ ( k ) + ∑ n = 0 k − 1 ψ ( 0 ) ( z + n k )

for the digamma function.

Series representation

The polygamma function has the series representation

ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ∑ k = 0 ∞ 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

ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ζ ( 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 Γ ( z ) = z e γ z ∏ n = 1 ∞ ( 1 + z n ) e − z n .

This is a result of the Weierstrass factorization theorem. Thus, the gamma function may now be defined as:

Γ ( z ) = e − γ z z ∏ n = 1 ∞ ( 1 + z n ) − 1 e z n .

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

ln ⁡ Γ ( z ) = − γ z − ln ⁡ ( z ) + ∑ n = 1 ∞ ( z n − ln ⁡ ( 1 + z n ) ) .

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

ψ ( n ) ( z ) = d n + 1 d z n + 1 ln ⁡ Γ ( z ) = − γ δ n 0 − ( − 1 ) n n ! z n + 1 + ∑ k = 1 ∞ ( 1 k δ n 0 − ( − 1 ) n n ! ( k + z ) n + 1 )

Where δn0 is the Kronecker delta.

Also the Lerch transcendent

Φ ( − 1 , m + 1 , z ) = ∑ k = 0 ∞ ( − 1 ) k ( z + k ) m + 1

can be denoted in terms of polygamma function

Φ ( − 1 , m + 1 , z ) = 1 ( − 2 ) m + 1 m ! ( ψ ( m ) ( z 2 ) − ψ ( m ) ( z + 1 2 ) )

Taylor series

The Taylor series at z = 1 is

ψ ( m ) ( z + 1 ) = ∑ k = 0 ∞ ( − 1 ) m + k + 1 ( m + k ) ! k ! ζ ( m + k + 1 ) z k m ≥ 1

and

ψ ( 0 ) ( z + 1 ) = − γ + ∑ k = 1 ∞ ( − 1 ) k + 1 ζ ( 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:

ψ ( m ) ( z ) ∼ ( − 1 ) m + 1 ∑ k = 0 ∞ ( k + m − 1 ) ! k ! B k z k + m m ≥ 1

and

ψ ( 0 ) ( z ) ∼ ln ⁡ ( z ) − ∑ k = 1 ∞ B k k z k

where we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.

References

Polygamma function Wikipedia
(Text) CC BY-SA