Meromorphic 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 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 In mathematics , the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1) th derivative of the logarithm of the gamma function :
ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . {\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).} Thus
ψ ( 0 ) ( z ) = ψ ( z ) = Γ ′ ( z ) Γ ( z ) {\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}} holds where ψ (z ) is the digamma function and Γ(z ) is the gamma function . They are holomorphic on C ∖ Z ≤ 0 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}} . 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 [ edit ] When m > 0 and Re z > 0 , the polygamma function equals
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 = ( − 1 ) m + 1 m ! ζ ( m + 1 , z ) {\displaystyle {\begin{aligned}\psi ^{(m)}(z)&=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-zt}}{1-e^{-t}}}\,\mathrm {d} t\\&=-\int _{0}^{1}{\frac {t^{z-1}}{1-t}}(\ln t)^{m}\,\mathrm {d} t\\&=(-1)^{m+1}m!\zeta (m+1,z)\end{aligned}}} where ζ ( s , q ) {\displaystyle \zeta (s,q)} is the Hurwitz zeta function .
This expresses the polygamma function as the Laplace transform of (−1)m +1 tm / 1 − e −t . 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 e −t / t .
Recurrence relation [ edit ] It satisfies the recurrence relation
ψ ( m ) ( z + 1 ) = ψ ( m ) ( z ) + ( − 1 ) m m ! z m + 1 {\displaystyle \psi ^{(m)}(z+1)=\psi ^{(m)}(z)+{\frac {(-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 {\displaystyle {\frac {\psi ^{(m)}(n)}{(-1)^{m+1}\,m!}}=\zeta (1+m)-\sum _{k=1}^{n-1}{\frac {1}{k^{m+1}}}=\sum _{k=n}^{\infty }{\frac {1}{k^{m+1}}}\qquad m\geq 1} and
ψ ( 0 ) ( n ) = − γ + ∑ k = 1 n − 1 1 k {\displaystyle \psi ^{(0)}(n)=-\gamma \ +\sum _{k=1}^{n-1}{\frac {1}{k}}} for all n ∈ N {\displaystyle n\in \mathbb {N} } , where γ {\displaystyle \gamma } is the Euler–Mascheroni constant . Like the log-gamma function, the polygamma functions can be generalized from the domain N {\displaystyle \mathbb {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 monotonicity on R + {\displaystyle \mathbb {R} ^{+}} is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on R + {\displaystyle \mathbb {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 [ edit ] ( − 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 ) {\displaystyle (-1)^{m}\psi ^{(m)}(1-z)-\psi ^{(m)}(z)=\pi {\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\cot {\pi z}=\pi ^{m+1}{\frac {P_{m}(\cos {\pi z})}{\sin ^{m+1}(\pi 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 ) ) . {\displaystyle {\begin{aligned}P_{0}(x)&=x\\P_{m+1}(x)&=-\left((m+1)xP_{m}(x)+\left(1-x^{2}\right)P'_{m}(x)\right).\end{aligned}}} Multiplication theorem [ edit ] The multiplication theorem gives
k m + 1 ψ ( m ) ( k z ) = ∑ n = 0 k − 1 ψ ( m ) ( z + n k ) m ≥ 1 {\displaystyle k^{m+1}\psi ^{(m)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m)}\left(z+{\frac {n}{k}}\right)\qquad m\geq 1} and
k ψ ( 0 ) ( k z ) = k ln k + ∑ n = 0 k − 1 ψ ( 0 ) ( z + n k ) {\displaystyle k\psi ^{(0)}(kz)=k\ln {k}+\sum _{n=0}^{k-1}\psi ^{(0)}\left(z+{\frac {n}{k}}\right)} for the digamma function .
Series representation [ edit ] The polygamma function has the series representation
ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ∑ k = 0 ∞ 1 ( z + k ) m + 1 {\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\,m!\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}} 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
ψ ( m ) ( z ) = ( − 1 ) m + 1 m ! ζ ( m + 1 , z ) . {\displaystyle \psi ^{(m)}(z)=(-1)^{m+1}\,m!\,\zeta (m+1,z).} This relation can for example be used to compute the special values[ 1]
ψ ( 2 n − 1 ) ( 1 4 ) = 4 2 n − 1 2 n ( π 2 n ( 2 2 n − 1 ) | B 2 n | + 2 ( 2 n ) ! β ( 2 n ) ) ; {\displaystyle \psi ^{(2n-1)}\left({\frac {1}{4}}\right)={\frac {4^{2n-1}}{2n}}\left(\pi ^{2n}(2^{2n}-1)|B_{2n}|+2(2n)!\beta (2n)\right);} ψ ( 2 n − 1 ) ( 3 4 ) = 4 2 n − 1 2 n ( π 2 n ( 2 2 n − 1 ) | B 2 n | − 2 ( 2 n ) ! β ( 2 n ) ) ; {\displaystyle \psi ^{(2n-1)}\left({\frac {3}{4}}\right)={\frac {4^{2n-1}}{2n}}\left(\pi ^{2n}(2^{2n}-1)|B_{2n}|-2(2n)!\beta (2n)\right);} ψ ( 2 n ) ( 1 4 ) = − 2 2 n − 1 ( π 2 n + 1 | E 2 n | + 2 ( 2 n ) ! ( 2 2 n + 1 − 1 ) ζ ( 2 n + 1 ) ) ; {\displaystyle \psi ^{(2n)}\left({\frac {1}{4}}\right)=-2^{2n-1}\left(\pi ^{2n+1}|E_{2n}|+2(2n)!(2^{2n+1}-1)\zeta (2n+1)\right);} ψ ( 2 n ) ( 3 4 ) = 2 2 n − 1 ( π 2 n + 1 | E 2 n | − 2 ( 2 n ) ! ( 2 2 n + 1 − 1 ) ζ ( 2 n + 1 ) ) . {\displaystyle \psi ^{(2n)}\left({\frac {3}{4}}\right)=2^{2n-1}\left(\pi ^{2n+1}|E_{2n}|-2(2n)!(2^{2n+1}-1)\zeta (2n+1)\right).} 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 . {\displaystyle {\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)e^{-{\frac {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 . {\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{\frac {z}{n}}.} Now, the natural logarithm of the gamma function is easily representable:
ln Γ ( z ) = − γ z − ln ( z ) + ∑ k = 1 ∞ ( z k − ln ( 1 + z k ) ) . {\displaystyle \ln \Gamma (z)=-\gamma z-\ln(z)+\sum _{k=1}^{\infty }\left({\frac {z}{k}}-\ln \left(1+{\frac {z}{k}}\right)\right).} 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 ) {\displaystyle \psi ^{(n)}(z)={\frac {\mathrm {d} ^{n+1}}{\mathrm {d} z^{n+1}}}\ln \Gamma (z)=-\gamma \delta _{n0}-{\frac {(-1)^{n}n!}{z^{n+1}}}+\sum _{k=1}^{\infty }\left({\frac {1}{k}}\delta _{n0}-{\frac {(-1)^{n}n!}{(k+z)^{n+1}}}\right)} Where δ n 0 is the Kronecker delta .
Also the Lerch transcendent
Φ ( − 1 , m + 1 , z ) = ∑ k = 0 ∞ ( − 1 ) k ( z + k ) m + 1 {\displaystyle \Phi (-1,m+1,z)=\sum _{k=0}^{\infty }{\frac {(-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 ) ) {\displaystyle \Phi (-1,m+1,z)={\frac {1}{(-2)^{m+1}m!}}\left(\psi ^{(m)}\left({\frac {z}{2}}\right)-\psi ^{(m)}\left({\frac {z+1}{2}}\right)\right)} 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 {\displaystyle \psi ^{(m)}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}{\frac {(m+k)!}{k!}}\zeta (m+k+1)z^{k}\qquad m\geq 1} and
ψ ( 0 ) ( z + 1 ) = − γ + ∑ k = 1 ∞ ( − 1 ) k + 1 ζ ( k + 1 ) z k {\displaystyle \psi ^{(0)}(z+1)=-\gamma +\sum _{k=1}^{\infty }(-1)^{k+1}\zeta (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 [ edit ] These non-converging series can be used to get quickly an approximation value with a certain numeric at-least-precision for large arguments:[ 2]
ψ ( m ) ( z ) ∼ ( − 1 ) m + 1 ∑ k = 0 ∞ ( k + m − 1 ) ! k ! B k z k + m m ≥ 1 {\displaystyle \psi ^{(m)}(z)\sim (-1)^{m+1}\sum _{k=0}^{\infty }{\frac {(k+m-1)!}{k!}}{\frac {B_{k}}{z^{k+m}}}\qquad m\geq 1} and
ψ ( 0 ) ( z ) ∼ ln ( z ) − ∑ k = 1 ∞ B k k z k {\displaystyle \psi ^{(0)}(z)\sim \ln(z)-\sum _{k=1}^{\infty }{\frac {B_{k}}{kz^{k}}}} where we have chosen B 1 = 1 / 2 , i.e. the Bernoulli numbers of the second kind.
The hyperbolic cotangent satisfies the inequality
t 2 coth t 2 ≥ 1 , {\displaystyle {\frac {t}{2}}\operatorname {coth} {\frac {t}{2}}\geq 1,} and this implies that the function
t m 1 − e − t − ( t m − 1 + t m 2 ) {\displaystyle {\frac {t^{m}}{1-e^{-t}}}-\left(t^{m-1}+{\frac {t^{m}}{2}}\right)} 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
( − 1 ) m + 1 ψ ( m ) ( x ) − ( ( m − 1 ) ! x m + m ! 2 x m + 1 ) {\displaystyle (-1)^{m+1}\psi ^{(m)}(x)-\left({\frac {(m-1)!}{x^{m}}}+{\frac {m!}{2x^{m+1}}}\right)} is completely monotone. The convexity inequality et ≥ 1 + t implies that
( t m − 1 + t m ) − t m 1 − e − t {\displaystyle \left(t^{m-1}+t^{m}\right)-{\frac {t^{m}}{1-e^{-t}}}} is non-negative for all m ≥ 1 and t ≥ 0 , so a similar Laplace transformation argument yields the complete monotonicity of
( ( m − 1 ) ! x m + m ! x m + 1 ) − ( − 1 ) m + 1 ψ ( m ) ( x ) . {\displaystyle \left({\frac {(m-1)!}{x^{m}}}+{\frac {m!}{x^{m+1}}}\right)-(-1)^{m+1}\psi ^{(m)}(x).} Therefore, for all m ≥ 1 and x > 0 ,
( m − 1 ) ! x m + m ! 2 x m + 1 ≤ ( − 1 ) m + 1 ψ ( m ) ( x ) ≤ ( m − 1 ) ! x m + m ! x m + 1 . {\displaystyle {\frac {(m-1)!}{x^{m}}}+{\frac {m!}{2x^{m+1}}}\leq (-1)^{m+1}\psi ^{(m)}(x)\leq {\frac {(m-1)!}{x^{m}}}+{\frac {m!}{x^{m+1}}}.} Since both bounds are strictly positive for x > 0 {\displaystyle x>0} , we have:
ln Γ ( x ) {\displaystyle \ln \Gamma (x)} is strictly convex . For m = 0 {\displaystyle m=0} , the digamma function, ψ ( x ) = ψ ( 0 ) ( x ) {\displaystyle \psi (x)=\psi ^{(0)}(x)} , is strictly monotonic increasing and strictly concave . For m {\displaystyle m} odd, the polygamma functions, ψ ( 1 ) , ψ ( 3 ) , ψ ( 5 ) , … {\displaystyle \psi ^{(1)},\psi ^{(3)},\psi ^{(5)},\ldots } , are strictly positive, strictly monotonic decreasing and strictly convex. For m {\displaystyle m} even the polygamma functions, ψ ( 2 ) , ψ ( 4 ) , ψ ( 6 ) , … {\displaystyle \psi ^{(2)},\psi ^{(4)},\psi ^{(6)},\ldots } , are strictly negative, strictly monotonic increasing and strictly concave. This can be seen in the first plot above.
Trigamma bounds and asymptote [ edit ] For the case of the trigamma function ( m = 1 {\displaystyle m=1} ) the final inequality formula above for x > 0 {\displaystyle x>0} , can be rewritten as:
x + 1 2 x 2 ≤ ψ ( 1 ) ( x ) ≤ x + 1 x 2 {\displaystyle {\frac {x+{\frac {1}{2}}}{x^{2}}}\leq \psi ^{(1)}(x)\leq {\frac {x+1}{x^{2}}}} so that for x ≫ 1 {\displaystyle x\gg 1} : ψ ( 1 ) ( x ) ≈ 1 x {\displaystyle \psi ^{(1)}(x)\approx {\frac {1}{x}}} .