# §33.19 Power-Series Expansions in $r$

 33.19.1 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=r^{\ell+1}\sum_{k=0}^{% \infty}\alpha_{k}r^{k},$

where

 33.19.2 $\displaystyle\alpha_{0}$ $\displaystyle=2^{\ell+1}/(2\ell+1)!,$ $\displaystyle\alpha_{1}$ $\displaystyle=-\alpha_{0}/(\ell+1),$ $\displaystyle k(k+2\ell+1)\alpha_{k}+2\alpha_{k-1}+\epsilon\alpha_{k-2}$ $\displaystyle=0,$ $k=2,3,\dots$. Defines: $\alpha_{k}$: term (locally) Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer, $\ell$: nonnegative integer and $\epsilon$: real parameter Permalink: http://dlmf.nist.gov/33.19.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 33.19
 33.19.3 $2\pi\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)={\sum_{k=0}^{2\ell}% \frac{(2\ell-k)!\gamma_{k}}{k!}(2r)^{k-\ell}-\sum_{k=0}^{\infty}\delta_{k}r^{k% +\ell+1}}-A(\epsilon,\ell)\left(2\mathop{\ln\/}\nolimits|2r/\kappa|+\Re{% \mathop{\psi\/}\nolimits\!\left(\ell+1+\kappa\right)}+\Re{\mathop{\psi\/}% \nolimits\!\left(-\ell+\kappa\right)}\right){\mathop{f\/}\nolimits\!\left(% \epsilon,\ell;r\right),}$ $r\neq 0$.

Here $\kappa$ is defined by (33.14.6), $A(\epsilon,\ell)$ is defined by (33.14.11) or (33.14.12), $\gamma_{0}=1$, $\gamma_{1}=1$, and

 33.19.4 $\gamma_{k}-\gamma_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\gamma_{k-2}=0,$ $k=2,3,\dots$. Symbols: $k$: nonnegative integer, $\ell$: nonnegative integer, $\epsilon$: real parameter and $\gamma_{k}$: term Permalink: http://dlmf.nist.gov/33.19.E4 Encodings: TeX, pMML, png See also: Annotations for 33.19

Also,

 33.19.5 $\displaystyle\delta_{0}$ $\displaystyle=\left(\beta_{2\ell+1}-2(\mathop{\psi\/}\nolimits\!\left(2\ell+2% \right)+\mathop{\psi\/}\nolimits\!\left(1\right))A(\epsilon,\ell)\right)\alpha% _{0},$ $\displaystyle\delta_{1}$ $\displaystyle=\left(\beta_{2\ell+2}-2(\mathop{\psi\/}\nolimits\!\left(2\ell+3% \right)+\mathop{\psi\/}\nolimits\!\left(2\right))A(\epsilon,\ell)\right)\alpha% _{1},$ Defines: $\delta_{k}$: term (locally) Symbols: $\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$: psi (or digamma) function, $k$: nonnegative integer, $\ell$: nonnegative integer, $\epsilon$: real parameter, $A(\epsilon,\ell)$: function, $\alpha_{k}$: term and $\beta_{k}$: term Permalink: http://dlmf.nist.gov/33.19.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 33.19
 33.19.6 $k(k+2\ell+1)\delta_{k}+2\delta_{k-1}+\epsilon\delta_{k-2}+2(2k+2\ell+1)A(% \epsilon,\ell)\alpha_{k}=0,$ $k=2,3,\dots$,

with $\beta_{0}=\beta_{1}=0$, and

 33.19.7 $\beta_{k}-\beta_{k-1}+\tfrac{1}{4}(k-1)(k-2\ell-2)\epsilon\beta_{k-2}+\tfrac{1% }{2}(k-1)\epsilon\gamma_{k-2}=0,$ $k=2,3,\dots$.

The expansions (33.19.1) and (33.19.3) converge for all finite values of $r$, except $r=0$ in the case of (33.19.3).