# §36.8 Convergent Series Expansions

 36.8.1 $\displaystyle\Psi_{K}\left(\mathbf{x}\right)$ $\displaystyle=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}\exp\left(i\dfrac{\pi(2n% +1)}{2(K+2)}\right)\Gamma\left(\dfrac{2n+1}{K+2}\right)a_{2n}(\mathbf{x}),$ $K$ even, $\displaystyle\Psi_{K}\left(\mathbf{x}\right)$ $\displaystyle=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}i^{n}\cos\left(\dfrac{% \pi(n(K+1)-1)}{2(K+2)}\right)\Gamma\left(\dfrac{n+1}{K+2}\right)a_{n}(\mathbf{% x}),$ $K$ odd,

where

 36.8.2 $\displaystyle a_{0}(\mathbf{x})$ $\displaystyle=1,$ $\displaystyle a_{n+1}(\mathbf{x})$ $\displaystyle=\dfrac{i}{n+1}\sum_{p=0}^{\min(n,K-1)}(p+1)x_{p+1}a_{n-p}(% \mathbf{x}),$ $n=0,1,2,\dots$. ⓘ Defines: $a_{n}(\mathbf{x})$: coefficients (locally) Symbols: $m$: integer, $K$: codimension and $x_{i}$: real parameter Permalink: http://dlmf.nist.gov/36.8.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 36.8 and 36

For multinomial power series for $\Psi_{K}\left(\mathbf{x}\right)$, see Connor and Curtis (1982).

 36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum\limits_{n=0}^{\infty}(-% 3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\mathrm{Ai}\left(x\right)\mathrm{% Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)d_% {n}(y)}{n!}+\mathrm{Ai}'\left(x\right)\mathrm{Ai}\left(y\right)\sum\limits_{n=% 2}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)c_{n}(y)}{n!}+\mathrm{Ai}'\left(x% \right)\mathrm{Ai}'\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)d_{n}(y)}{n!},$

and

 36.8.4 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$

where

 36.8.5 $f_{n}(\zeta,\zeta^{\ast})=c_{n}(\zeta)c_{n}(\zeta^{\ast})\mathrm{Ai}\left(% \zeta\right)\mathrm{Bi}\left(\zeta^{\ast}\right)+c_{n}(\zeta)d_{n}(\zeta^{\ast% })\mathrm{Ai}\left(\zeta\right)\mathrm{Bi}'\left(\zeta^{\ast}\right)+d_{n}(% \zeta)c_{n}(\zeta^{\ast})\mathrm{Ai}'\left(\zeta\right)\mathrm{Bi}\left(\zeta^% {\ast}\right)+d_{n}(\zeta)d_{n}(\zeta^{\ast})\mathrm{Ai}'\left(\zeta\right)% \mathrm{Bi}'\left(\zeta^{\ast}\right),$ ⓘ Defines: $f_{n}$: function (locally) Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $m$: integer, $c_{n}(t)$: coefficients and $d_{n}(t)$: coefficients Permalink: http://dlmf.nist.gov/36.8.E5 Encodings: TeX, pMML, png See also: Annotations for 36.8 and 36

with asterisks denoting complex conjugates, and

 36.8.6 $\displaystyle c_{0}(t)$ $\displaystyle=1,$ $\displaystyle d_{0}(t)$ $\displaystyle=0,$ $\displaystyle c_{n+1}(t)$ $\displaystyle=c_{n}^{\mspace{1.0mu }\prime}(t)+td_{n}(t),$ $\displaystyle d_{n+1}(t)$ $\displaystyle=c_{n}(t)+d_{n}^{\mspace{1.0mu }\prime}(t).$ ⓘ Defines: $c_{n}(t)$: coefficients (locally) and $d_{n}(t)$: coefficients (locally) Symbols: $m$: integer and $t$: variable Permalink: http://dlmf.nist.gov/36.8.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 36.8 and 36