# §36.8 Convergent Series Expansions

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

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

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

and

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

where

 36.8.5 $f_{n}(\zeta,\zeta^{\ast})=c_{n}(\zeta)c_{n}(\zeta^{\ast})\mathop{\mathrm{Ai}\/% }\nolimits\!\left(\zeta\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\zeta^{% \ast}\right)+c_{n}(\zeta)d_{n}(\zeta^{\ast})\mathop{\mathrm{Ai}\/}\nolimits\!% \left(\zeta\right)\mathop{\mathrm{Bi}\/}\nolimits'\!\left(\zeta^{\ast}\right)+% d_{n}(\zeta)c_{n}(\zeta^{\ast})\mathop{\mathrm{Ai}\/}\nolimits'\!\left(\zeta% \right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\zeta^{\ast}\right)+d_{n}(\zeta)% d_{n}(\zeta^{\ast})\mathop{\mathrm{Ai}\/}\nolimits'\!\left(\zeta\right)\mathop% {\mathrm{Bi}\/}\nolimits'\!\left(\zeta^{\ast}\right),$ Defines: $f_{n}$: function (locally) Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: 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

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