# Euler transformation

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##### 2: 3.9 Acceleration of Convergence
###### §3.9(ii) Euler’s Transformation of Series
Euler’s transformation is usually applied to alternating series. …
##### 3: 2.11 Remainder Terms; Stokes Phenomenon
The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. A simple example is provided by Euler’s transformation3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). …
##### 4: 2.5 Mellin Transform Methods
Since $\mathscr{M}\mskip-3.0mue^{-t}\mskip 3.0mu\left(z\right)=\Gamma\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_{1}$ and $p_{2}$ such that $-c, $p_{2}<\min(1,\Re\beta_{0})$, and
2.5.40 $I_{j}(x)=\frac{1}{2\pi i}\int_{p_{j}-i\infty}^{p_{j}+i\infty}x^{-z}\Gamma\left% (1-z\right)\mathscr{M}\mskip-3.0muh_{j}\mskip 3.0mu\left(z\right)\mathrm{d}z,$ $j=1,2$.
2.5.41 $I_{1}(x)=\mathscr{M}\mskip-3.0muh_{1}\mskip 3.0mu\left(1\right)x^{-1}+\frac{1}% {2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)\mathscr% {M}\mskip-3.0muh_{1}\mskip 3.0mu\left(z\right)\mathrm{d}z,$
2.5.42 $I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue\left[-x^{-z}\Gamma\left(1-% z\right)\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)\right]+\frac{1}% {2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)\mathscr% {M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)\mathrm{d}z.$
2.5.47 $\Residue_{z=1}\left[-\zeta^{z-1}\Gamma\left(1-z\right)\mathscr{M}\mskip-3.0muh% _{2}\mskip 3.0mu\left(z\right)\right]=\left(-\ln\zeta-\gamma\right)-\mathscr{M% }\mskip-3.0muh_{1}\mskip 3.0mu\left(1\right),$
##### 5: 15.14 Integrals
15.14.1 $\int_{0}^{\infty}x^{s-1}\mathbf{F}\left({a,b\atop c};-x\right)\mathrm{d}x=% \frac{\Gamma\left(s\right)\Gamma\left(a-s\right)\Gamma\left(b-s\right)}{\Gamma% \left(a\right)\Gamma\left(b\right)\Gamma\left(c-s\right)},$ $\min(\Re a,\Re b)>\Re s>0$.
##### 6: 13.23 Integrals
13.23.1 $\int_{0}^{\infty}e^{-zt}t^{\nu-1}M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac% {\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)}{\left(z+\frac{1}{2}\right)^{\mu+\nu+% \frac{1}{2}}}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}+\mu-\kappa,\tfrac{1}{2}+\mu+% \nu\atop 1+2\mu};\frac{1}{z+\frac{1}{2}}\right),$ $\Re\mu+\nu+\tfrac{1}{2}>0$, $\Re z>\tfrac{1}{2}$.
13.23.2 $\int_{0}^{\infty}e^{-zt}t^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(t\right)\mathrm% {d}t=\Gamma\left(2\mu+1\right)\left(z+\tfrac{1}{2}\right)^{-\kappa-\mu-\frac{1% }{2}}\*\left(z-\tfrac{1}{2}\right)^{\kappa-\mu-\frac{1}{2}},$ $\Re\mu>-\tfrac{1}{2}$, $\Re z>\tfrac{1}{2}$,
13.23.3 $\frac{1}{\Gamma\left(1+2\mu\right)}\int_{0}^{\infty}e^{-\frac{1}{2}t}t^{\nu-1}% M_{\kappa,\mu}\left(t\right)\mathrm{d}t=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}% \right)\Gamma\left(\kappa-\nu\right)}{\Gamma\left(\frac{1}{2}+\mu+\kappa\right% )\Gamma\left(\frac{1}{2}+\mu-\nu\right)},$ $-\tfrac{1}{2}-\Re\mu<\Re\nu<\Re\kappa$.
13.23.6 $\frac{1}{\Gamma\left(1+2\mu\right)2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+% \frac{1}{2}t^{-1}}t^{\kappa}M_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac% {z^{-\kappa-\frac{1}{2}}}{\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}I_{2\mu}% \left(2\sqrt{z}\right),$ $\Re z>0$.
13.23.7 $\frac{1}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt+\frac{1}{2}t^{-1}}t^{\kappa% }W_{\kappa,\mu}\left(t^{-1}\right)\mathrm{d}t=\frac{2z^{-\kappa-\frac{1}{2}}}{% \Gamma\left(\frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa% \right)}K_{2\mu}\left(2\sqrt{z}\right),$ $\Re z>0$.
##### 7: 11.7 Integrals and Sums
11.7.10 $\int_{0}^{\infty}t^{-\nu-1}\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=\frac{\pi% }{2^{\nu+1}\Gamma\left(\nu+1\right)},$ $\Re\nu>-\tfrac{3}{2}$,
11.7.11 $\int_{0}^{\infty}t^{\mu-\nu-1}\mathbf{H}_{\nu}\left(t\right)\mathrm{d}t=\frac{% \Gamma\left(\tfrac{1}{2}\mu\right)2^{\mu-\nu-1}\tan\left(\tfrac{1}{2}\pi\mu% \right)}{\Gamma\left(\nu-\tfrac{1}{2}\mu+1\right)},$ $|\Re\mu|<1$, $\Re\nu>\Re\mu-\tfrac{3}{2}$,
11.7.12 $\int_{0}^{\infty}t^{-\mu-\nu}\mathbf{H}_{\mu}\left(t\right)\mathbf{H}_{\nu}% \left(t\right)\mathrm{d}t=\frac{\sqrt{\pi}\Gamma\left(\mu+\nu\right)}{2^{\mu+% \nu}\Gamma\left(\mu+\nu+\tfrac{1}{2}\right)\Gamma\left(\mu+\tfrac{1}{2}\right)% \Gamma\left(\nu+\tfrac{1}{2}\right)},$ $\Re\left(\mu+\nu\right)>0$.
##### 8: Bibliography M
• A. R. Miller and R. B. Paris (2011) Euler-type transformations for the generalized hypergeometric function ${}_{r+2}F_{r+1}(x)$ . Z. Angew. Math. Phys. 62 (1), pp. 31–45.
• ##### 9: 13.4 Integral Representations
13.4.16 ${\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(-t\right)}{\Gamma\left(b+t\right)}z^{t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,
13.4.17 $U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{% \mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)\Gamma% \left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a-b\right)}z^{-t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{3}{2}\pi$,
13.4.18 $U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty% }^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t\right)\Gamma\left(t\right)}{\Gamma% \left(a+t\right)}z^{-t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,
##### 10: 13.16 Integral Representations
13.16.10 $\frac{1}{\Gamma\left(1+2\mu\right)}M_{\kappa,\mu}\left(e^{\pm\pi\mathrm{i}}z% \right)=\frac{e^{\frac{1}{2}z\pm(\frac{1}{2}+\mu)\pi\mathrm{i}}}{2\pi\mathrm{i% }\Gamma\left(\frac{1}{2}+\mu-\kappa\right)}\*\int_{-\mathrm{i}\infty}^{\mathrm% {i}\infty}\frac{\Gamma\left(t-\kappa\right)\Gamma\left(\frac{1}{2}+\mu-t\right% )}{\Gamma\left(\frac{1}{2}+\mu+t\right)}z^{t}\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,
13.16.11 $W_{\kappa,\mu}\left(z\right)=\frac{e^{-\frac{1}{2}z}}{2\pi\mathrm{i}}\*\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)\Gamma\left(-\kappa-t\right)}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)\Gamma\left(\frac{1}{2}-\mu-\kappa\right)}z^{-t}% \mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{3}{2}\pi$,
13.16.12 $W_{\kappa,\mu}\left(z\right)=\frac{e^{\frac{1}{2}z}}{2\pi\mathrm{i}}\int_{-% \mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(\frac{1}{2}+\mu+t\right)% \Gamma\left(\frac{1}{2}-\mu+t\right)}{\Gamma\left(1-\kappa+t\right)}z^{-t}% \mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$,