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Balanced polygamma function
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<h2 id="definition">Definition</h2> <p>The generalized polygamma function is defined as follows: </p> ψ ( z , q ) = ζ ′ ( z + 1 , q ) + ( ψ ( − z ) + γ ) ζ ( z + 1 , q ) Γ ( − z ) {\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\bigl (}\psi (-z)+\gamma {\bigr )}\zeta (z+1,q)}{\Gamma (-z)}}} <p>or alternatively, </p> ψ ( z , q ) = e − γ z ∂ ∂ z ( e γ z ζ ( z + 1 , q ) Γ ( − z ) ) , {\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right),} <p>where <i>ψ</i>(<i>z</i>) is the <a href="/facts/Polygamma_function/IF6W68aL">polygamma function</a> and <i>ζ</i>(<i>z</i>,<i>q</i>), is the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a>. </p><p>The function is balanced, in that it satisfies the conditions </p> f ( 0 ) = f ( 1 ) and ∫ 0 1 f ( x ) d x = 0 {\displaystyle f(0)=f(1)\quad {\text{and}}\quad \int _{0}^{1}f(x)\,dx=0} . <h2 id="relations">Relations</h2> <p>Several special functions can be expressed in terms of generalized polygamma function. </p> ψ ( x ) = ψ ( 0 , x ) ψ ( n ) ( x ) = ψ ( n , x ) n ∈ N Γ ( x ) = exp ( ψ ( − 1 , x ) + 1 2 ln 2 π ) ζ ( z , q ) = ( − 1 ) z Γ ( z ) ψ ( z − 1 , q ) ζ ′ ( − 1 , x ) = ψ ( − 2 , x ) + x 2 2 − x 2 + 1 12 {\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\ln 2\pi \right)\\\zeta (z,q)&={\frac {(-1)^{z}}{\Gamma (z)}}\psi (z-1,q)\\\zeta '(-1,x)&=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}\\\end{aligned}}} K ( z ) = A exp ( ψ ( − 2 , z ) + z 2 − z 2 ) {\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)} <p>where <i>K</i>(<i>z</i>) is the <a href="/facts/K-function/XVxp0lgJ">K-function</a> and A is the <a href="/facts/Glaisher_constant/lK8noN4A">Glaisher constant</a>. </p> <h2 id="special-values">Special values</h2> <p>The balanced polygamma function can be expressed in a closed form at certain points (where A is the <a href="/facts/Glaisher_constant/lK8noN4A">Glaisher constant</a> and G is the <a href="/facts/Catalan_constant/c9fNqG8j">Catalan constant</a>): </p> ψ ( − 2 , 1 4 ) = 1 8 ln A + G 4 π ψ ( − 2 , 1 2 ) = 1 2 ln A − 1 24 ln 2 ψ ( − 3 , 1 2 ) = 3 ζ ( 3 ) 32 π 2 ψ ( − 2 , 1 ) = − ln A ψ ( − 3 , 1 ) = − ζ ( 3 ) 8 π 2 ψ ( − 2 , 2 ) = − ln A − 1 ψ ( − 3 , 2 ) = − ζ ( 3 ) 8 π 2 − 3 4 {\displaystyle {\begin{aligned}\psi \left(-2,{\tfrac {1}{4}}\right)&={\tfrac {1}{8}}\ln A+{\frac {G}{4\pi }}&&\\\psi \left(-2,{\tfrac {1}{2}}\right)&={\tfrac {1}{2}}\ln A-{\tfrac {1}{24}}\ln 2&\\\psi \left(-3,{\tfrac {1}{2}}\right)&={\frac {3\zeta (3)}{32\pi ^{2}}}\\\psi (-2,1)&=-\ln A&\\\psi (-3,1)&={\frac {-\zeta (3)}{8\pi ^{2}}}\\\psi (-2,2)&=-\ln A-1&\\\psi (-3,2)&={\frac {-\zeta (3)}{8\pi ^{2}}}-{\tfrac {3}{4}}\\\end{aligned}}} <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Espinosa, Olivier; Moll, Victor Hugo (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. doi:10.1080/10652460310001600573. <a href="/wiki/Victor_Hugo_Moll" target="_blank">/wiki/Victor_Hugo_Moll</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>