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Multivariate gamma function
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<h2 id="derivatives">Derivatives</h2> <p>We may define the multivariate <a href="/facts/Digamma_function/nDTNu86e">digamma function</a> as </p> ψ p ( a ) = ∂ log Γ p ( a ) ∂ a = ∑ i = 1 p ψ ( a + ( 1 − i ) / 2 ) , {\displaystyle \psi _{p}(a)={\frac {\partial \log \Gamma _{p}(a)}{\partial a}}=\sum _{i=1}^{p}\psi (a+(1-i)/2),} <p>and the general <a href="/facts/Polygamma_function/IF6W68aL">polygamma function</a> as </p> ψ p ( n ) ( a ) = ∂ n log Γ p ( a ) ∂ a n = ∑ i = 1 p ψ ( n ) ( a + ( 1 − i ) / 2 ) . {\displaystyle \psi _{p}^{(n)}(a)={\frac {\partial ^{n}\log \Gamma _{p}(a)}{\partial a^{n}}}=\sum _{i=1}^{p}\psi ^{(n)}(a+(1-i)/2).} <h3>Calculation steps</h3> <ul><li>Since</li></ul> Γ p ( a ) = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( a + 1 − j 2 ) , {\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right),} it follows that ∂ Γ p ( a ) ∂ a = π p ( p − 1 ) / 4 ∑ i = 1 p ∂ Γ ( a + 1 − i 2 ) ∂ a ∏ j = 1 , j ≠ i p Γ ( a + 1 − j 2 ) . {\displaystyle {\frac {\partial \Gamma _{p}(a)}{\partial a}}=\pi ^{p(p-1)/4}\sum _{i=1}^{p}{\frac {\partial \Gamma \left(a+{\frac {1-i}{2}}\right)}{\partial a}}\prod _{j=1,j\neq i}^{p}\Gamma \left(a+{\frac {1-j}{2}}\right).} <ul><li>By definition of the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>, ψ,</li></ul> ∂ Γ ( a + ( 1 − i ) / 2 ) ∂ a = ψ ( a + ( i − 1 ) / 2 ) Γ ( a + ( i − 1 ) / 2 ) {\displaystyle {\frac {\partial \Gamma (a+(1-i)/2)}{\partial a}}=\psi (a+(i-1)/2)\Gamma (a+(i-1)/2)} it follows that ∂ Γ p ( a ) ∂ a = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( a + ( 1 − j ) / 2 ) ∑ i = 1 p ψ ( a + ( 1 − i ) / 2 ) = Γ p ( a ) ∑ i = 1 p ψ ( a + ( 1 − i ) / 2 ) . {\displaystyle {\begin{aligned}{\frac {\partial \Gamma _{p}(a)}{\partial a}}&=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2)\sum _{i=1}^{p}\psi (a+(1-i)/2)\\[4pt]&=\Gamma _{p}(a)\sum _{i=1}^{p}\psi (a+(1-i)/2).\end{aligned}}} <ul><li>1. James, A. (1964). <a href="https://doi.org/10.1214%2Faoms%2F1177703550">"Distributions of Matrix Variates and Latent Roots Derived from Normal Samples"</a>. <i><a href="/facts/Annals_of_Mathematical_Statistics/R8H9bM4d">Annals of Mathematical Statistics</a></i>. 35 (2): 475–501. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faoms%2F1177703550">10.1214/aoms/1177703550</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0181057">0181057</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0121.36605">0121.36605</a>.</li> <li>2. A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>James, Alan T. (June 1964). "Distributions of Matrix Variates and Latent Roots Derived from Normal Samples". The Annals of Mathematical Statistics. 35 (2): 475–501. doi:10.1214/aoms/1177703550. ISSN 0003-4851. <a href="http://projecteuclid.org/euclid.aoms/1177703550" target="_blank">http://projecteuclid.org/euclid.aoms/1177703550</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Anderson, T W (1984). An Introduction to Multivariate Statistical Analysis. New York: John Wiley and Sons. pp. Ch. 7. ISBN 0-471-88987-3. <a href="0-471-88987-3" target="_blank">0-471-88987-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>D. St. P. Richards (n.d.). "Chapter 35 Functions of Matrix Argument". Digital Library of Mathematical Functions. Retrieved 23 May 2022. <a href="/wiki/Donald_Richards_(statistician)" target="_blank">/wiki/Donald_Richards_(statistician)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>