Reciprocal times Derivative of Gamma Function/Corollary 2

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Corollary to Reciprocal times Derivative of Gamma Function

Let $a$ and $b \in \C$ such that $\paren {\dfrac a {2 b} + 1} \in \C \setminus \Z_{\le 0}$ and $\paren {\dfrac a {2 b} + \dfrac 1 2 } \in \C \setminus \Z_{\le 0}$

Then:

$\ds \map \psi {\dfrac a {2 b} + 1} - \map \psi {\dfrac a {2 b} + \dfrac 1 2} = 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k} $

where:

$\psi$ is the digamma function


Proof

\(\ds \map \psi {\frac a {2 b} + 1}\) \(=\) \(\ds \dfrac {\map {\Gamma'} {\frac a {2 b} + 1} } {\map \Gamma {\frac a {2 b} + 1} }\) Definition of Digamma Function: $\Gamma$ denotes the gamma function
\(\ds \) \(=\) \(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {n + \dfrac a {2 b} } }\) Reciprocal times Derivative of Gamma Function
\(\ds \) \(=\) \(\ds -\gamma + \frac {2 b} {2 b} \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {n + \dfrac a {2 b} } }\) multiplying top and bottom by $2 b$
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds -\gamma + 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {2 b n} - \frac 1 {2 b n + a} }\)
\(\ds \map \psi {\paren {\frac a {2 b} - \frac 1 2} + 1}\) \(=\) \(\ds \dfrac {\map {\Gamma'} {\paren {\frac a {2 b} - \frac 1 2} + 1} } {\map \Gamma {\paren {\frac a {2 b} - \frac 1 2} + 1} }\) Definition of Digamma Function: $\Gamma$ denotes the gamma function
\(\ds \) \(=\) \(\ds -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {n + \paren {\dfrac a {2 b} - \dfrac 1 2} } }\) Reciprocal times Derivative of Gamma Function
\(\ds \) \(=\) \(\ds -\gamma + \frac {2 b} {2 b} \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {n + \paren {\dfrac a {2 b} - \dfrac 1 2} } }\) multiplying top and bottom by $2 b$
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds -\gamma + 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {2 b n} - \frac 1 {2 b n + a - b} }\)
\(\ds \leadsto \ \ \) \(\ds \map \psi {\frac a {2 b} + 1} - \map \psi {\frac a {2 b} + \frac 1 2 }\) \(=\) \(\ds \paren {-\gamma + 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {2 b n} - \frac 1 {2 b n + a} } } - \paren {-\gamma + 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {2 b n} - \frac 1 {2 b n + a - b} } }\) subtracting $2$ from $1$
\(\ds \) \(=\) \(\ds 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {2 b n} - \frac 1 {2 b n} } + \paren {\frac 1 {2 b n + a - b} - \frac 1 {2 b n + a} }\) Linear Combination of Convergent Series
\(\ds \) \(=\) \(\ds 2 b \sum_{n \mathop = 1}^\infty \paren {\frac 1 {b \paren {2 n - 1} + a} - \frac 1 {2 b n + a} }\)
\(\ds \) \(=\) \(\ds 2 b \paren {\paren {\frac 1 {a + b} - \frac 1 {a + 2 b} } + \paren {\frac 1 {a + 3 b} - \frac 1 {a + 4 b} } + \paren {\frac 1 {a + 5 b} - \frac 1 {a + 6 b} } + \cdots}\)
\(\ds \) \(=\) \(\ds 2 b \sum_{k \mathop = 1}^\infty \dfrac {\paren {-1}^{k + 1} } {a + b k}\)

$\blacksquare$


Sources

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