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In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln ⁡...

have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. When m > 0 and Re z > 0, the polygamma function equals ψ ( m...

\operatorname {cov} [\ln X,\ln(1-X)]=-\psi _{1}(\alpha +\beta )} where the trigamma function, denoted ψ1(α), is the second of the polygamma functions, and...

X]=\psi ^{(1)}(\alpha )} where ψ ( 1 ) {\displaystyle \psi ^{(1)}} is the trigamma function. This can be derived using the exponential family formula for...

special cases the digamma function ψ ( z ) {\displaystyle \psi (z)} the trigamma function ψ 1 ( z ) {\displaystyle \psi ^{1}(z)} and in physics to the quantum...

G appears in values of the second polygamma function, also called the trigamma function, at fractional arguments: ψ 1 ( 1 4 ) = π 2 + 8 G ψ 1 ( 3 4 )...

{\displaystyle \psi } is the digamma function, ψ ′ {\displaystyle \psi '} is the trigamma function, and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta...

{3}}{3}}\left({\frac {\psi _{1}(1/3)}{2}}-{\frac {\pi ^{2}}{3}}\right)} with the trigamma function ψ 1 {\displaystyle \psi _{1}} . 1912 ? ? ✓ Bernstein's constant...

approximation Spouge, John L. (1994). "Computation of the Gamma, Digamma, and Trigamma Functions". SIAM Journal on Numerical Analysis. 31 (3): 931–000. doi:10...

\xi } only through the polygamma function of order 1 (also called the trigamma function): Var ⁡ [ Y ] = { ψ ′ ( 1 ) − ψ ′ ( − 1 / ξ + 1 ) for  ξ < 0 ...

the log-likelihood is taken for finding the maxima. Polygamma function Trigamma function Chebyshev expansions of the digamma function in Wimp, Jet (1961)...

solution can be written in terms of the digamma ψ {\displaystyle \psi } and trigamma ψ ′ {\displaystyle \psi '} functions: μ i ∗ = ψ ( α i ) − ψ ( α D ) , i...

_{1}\left({\frac {n+1-i}{2}}\right)} where ψ 1 {\displaystyle \psi _{1}} is the trigamma function. This comes up when computing the Fisher information of the Wishart...

\chi _{-4})} Another closed form expression may be given in terms of the trigamma function: V = 3 3 ( ψ 1 ( 1 / 3 ) 2 − π 2 3 ) {\displaystyle V={\frac {\sqrt...

_{m=0}^{\infty }B_{m}^{+}z^{m}} is an asymptotic series. It contains the trigamma function ψ1. From the generating functions above, one can obtain the following...

{1}{6n^{3}}}+{\frac {1}{30n^{5}}}-{\frac {1}{42n^{7}}}+\cdots .} This is related to the trigamma function: ∑ k = 1 n 1 k 2 = π 2 6 − ψ ( 1 ) ( n + 1 ) {\displaystyle \sum...

{\displaystyle \psi } and ψ ′ {\displaystyle \psi '} are the digamma function and trigamma function respectively. Given a value for ⁠ β {\displaystyle \textstyle...

1 ) ( x ) {\displaystyle \psi ^{\left(1\right)}\left(x\right)} is the trigamma function. The inverse gamma function also has the following asymptotic...

{\displaystyle \psi '=\psi ^{(1)}} is its first derivative, also known as the trigamma function, or the first polygamma function. Since ψ {\displaystyle \psi...

ISSN 1464-3839. Pairman, Eleanor. (1919). Tables of the digamma and trigamma functions. Cambridge University Press. OCLC 681090033. Pairman, Eleanor;...

( ν ) {\displaystyle \digamma ^{[1]}(\nu )} are values of digamma and trigamma functions at ν {\displaystyle \nu } , respectively. Demirhan and Hamurkaroglu...

for the Riemann zeta-function, Ψ is the digamma function, and Ψ1 is the trigamma function; see respectively eq. (43), (47) and (48) in Adamchik for the...