Catalan constant

Named after its inventor, E.Ch. Catalan (1814–1894), the Catalan constant $G$ (which is denoted also by $\lambda$) is defined by

\begin{equation} \tag{a1} G : = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { ( 2 k + 1 ) ^ { 2 } } \cong \end{equation}

\begin{equation*} \cong 0.915965594177219015 \ldots . \end{equation*}

If, in terms of the Digamma (or Psi) function $\psi ( z )$, defined by

\begin{equation} \tag{a2} \psi ( z ) : = \frac { d } { d z } \{ \operatorname { log } \Gamma ( z ) \} = \frac { \Gamma ^ { \prime } ( z ) } { \Gamma ( z ) } \end{equation}

or

\begin{equation*} \operatorname { log } \Gamma ( z ) = \int _ { 1 } ^ { z } \psi ( t ) d t, \end{equation*}

one puts

\begin{equation} \tag{a3} \beta ( z ) : = \frac { 1 } { 2 } \left[ \psi \left( \frac { 1 } { 2 } z + \frac { 1 } { 2 } \right) - \psi \left( \frac { 1 } { 2 } z \right) \right] = \end{equation}

\begin{equation*} = \sum _ { k = 0 } ^ { \infty } \frac { ( - 1 ) ^ { k } } { z + k }, \end{equation*}

where

\begin{equation*} z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 }^- , \quad \mathbf{Z} _ { 0 } ^ { - } : = \{ 0 , - 1 , - 2 , \ldots \}, \end{equation*}

then

\begin{equation} \tag{a4} G = - \frac { 1 } { 4 } \beta ^ { \prime } \left( \frac { 1 } { 2 } \right) \end{equation}

which provides a relationship between the Catalan constant $G$ and the Digamma function $\psi ( z )$.

The Catalan constant $G$ is related also to other functions, such as the Clausen function $\operatorname{Cl} _ { 2 } ( z )$, defined by

\begin{equation} \tag{a5} \operatorname {Cl} _ { 2 } ( z ) : = - \int _ { 0 } ^ { z } \operatorname { log } \left| 2 \operatorname { sin } \left( \frac { 1 } { 2 } t \right) \right| d t = \end{equation}

\begin{equation*} = \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { sin } ( k z ) } { k ^ { 2 } }, \end{equation*}

and the Hurwitz zeta function $\zeta ( s , a )$, which is defined, when $\operatorname { Re } s > 1$, by

\begin{equation} \tag{a6} \zeta ( s , a ) : = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { ( k + a ) ^ { s } }, \end{equation}

\begin{equation*} \operatorname { Re } s > 1 , a \in \mathbf{C} \backslash \mathbf{Z} ^{ - } _ { 0 }. \end{equation*}

Thus,

\begin{equation} \tag{a7} G = \operatorname{Cl} _ { 2 } ( \frac { 1 } { 2 } \pi ) = - \operatorname{Cl} _ { 2 } \left( \frac { 3 } { 2 } \pi \right) = \end{equation}

\begin{equation*} = \frac { 1 } { 16 } \left[ \zeta \left( 2 , \frac { 1 } { 4 } \right) - \zeta \left( 2 , \frac { 3 } { 4 } \right) \right]. \end{equation*}

Since

\begin{equation} \tag{a8} \psi ^ { ( n ) } ( z ) = ( - 1 ) ^ { n + 1 } n ! \zeta ( n + 1 , z ), \end{equation}

\begin{equation*} n \in \mathbf{N} : = \{ 1,2 , \ldots \} , z \in \mathbf{C} \backslash \mathbf{Z} _ { 0 } ^ { - }, \end{equation*}

the last expression in (a7) would follow also from (a4) in light of the definition in (a3).

A fairly large number of integrals and series can be evaluated in terms of the Catalan constant $G$. For example,

\begin{equation} \tag{a9} \int _ { 0 } ^ { 1 } \frac { t\operatorname { log } ( t ^ { - 1 } \pm t ) } { 1 + t ^ { 4 } } d t = \end{equation}

\begin{equation*} = \int _ { 1 } ^ { \infty } \frac { t \operatorname { log } ( t \pm t ^ { - 1 } ) } { 1 + t ^ { 4 } } d t = \frac { \pi } { 16 } \operatorname { log } 2 \pm \frac { G } { 4 }, \end{equation*}

\begin{equation} \tag{a10} \sum _ { k = 1 } ^ { \infty } \left( \frac { ( 2 k + 1 ) ! } { k ! ( k + 1 ) ! } \right) ^ { 2 } \frac { 2 ^ { - 4 k } } { k } = \end{equation}

\begin{equation*} = 4 \operatorname { log } 2 + 2 - \frac { 4 } { \pi } ( 2 G + 1 ), \end{equation*}

and

\begin{equation} \tag{a11} \sum _ { k = 1 } ^ { \infty } \frac { \zeta ( 2 k ) } { k ( 2 k + 1 ) 2 ^ { 4 k } } = \operatorname { log } ( \frac { \pi } { 2 } ) - 1 + \frac { 2 G } { \pi }, \end{equation}

where $\zeta ( s ) = \zeta ( s , 1 )$ denotes the familiar Riemann zeta-function.

References

[Fi]	Steven R. Finch, "Mathematical constants" , Encyclopedia of mathematics and its applications 94, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001