# Euler sums (first, second, third)

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##### 2: 10.2 Definitions
###### Bessel Functions of the Third Kind (Hankel Functions)
These solutions of (10.2.1) are denoted by ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$, and their defining properties are given by … The principal branches of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
##### 3: 10.8 Power Series
When $\nu$ is not an integer the corresponding expansions for $Y_{\nu}\left(z\right)$, ${H^{(1)}_{\nu}}\left(z\right)$, and ${H^{(2)}_{\nu}}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). … where $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$5.2(i)). …
10.8.2 $Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)+\gamma% \right)J_{0}\left(z\right)+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2% }}-(1+\tfrac{1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+% \tfrac{1}{3})\frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\dotsi\right),$
where $\gamma$ is Euler’s constant (§5.2(ii)). … The corresponding results for ${H^{(1)}_{n}}\left(z\right)$ and ${H^{(2)}_{n}}\left(z\right)$ are obtained via (10.4.3) with $\nu=n$. …
##### 4: 16.13 Appell Functions
16.13.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}{\left(\beta^{% \prime}\right)_{n}}}{{\left(\gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$,
16.13.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m}}% {\left(\beta^{\prime}\right)_{n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{% \prime}\right)_{n}}m!n!}x^{m}y^{n},$ $|x|+|y|<1$,
16.13.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m}}{\left(\alpha^{\prime}% \right)_{n}}{\left(\beta\right)_{m}}{\left(\beta^{\prime}\right)_{n}}}{{\left(% \gamma\right)_{m+n}}m!n!}x^{m}y^{n},$ $\max\left(|x|,|y|\right)<1$,
16.13.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta\right)_{m+n}}}{{\left(% \gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n!}x^{m}y^{n},$ $\sqrt{|x|}+\sqrt{|y|}<1$.
Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and $\gamma^{\prime}$ is a nonpositive integer. …
##### 5: 19.5 Maclaurin and Related Expansions
where ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$ is an Appell function (§16.13). … Coefficients of terms up to $\lambda^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\leq k^{2}\leq 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product … Series expansions of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ are surveyed and improved in Van de Vel (1969), and the case of $F\left(\phi,k\right)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\Pi\left(\phi,\alpha^{2},k\right)$ when $|\alpha^{2}|<1$ see Erdélyi et al. (1953b, §13.6(9)). …
##### 6: 10.43 Integrals
10.43.5 $\int_{x}^{\infty}\frac{K_{0}\left(t\right)}{t}\mathrm{d}t=\frac{1}{2}\left(\ln% \left(\tfrac{1}{2}x\right)+\gamma\right)^{2}+\frac{\pi^{2}}{24}-\sum_{k=1}^{% \infty}\left(\psi\left(k+1\right)+\frac{1}{2k}-\ln\left(\tfrac{1}{2}x\right)% \right)\frac{(\tfrac{1}{2}x)^{2k}}{2k(k!)^{2}},$
where $\psi=\ifrac{\Gamma'}{\Gamma}$ and $\gamma$ is Euler’s constant (§5.2). … For the second equation there is a cut in the $a$-plane along the interval $[0,1]$, and all quantities assume their principal values (§4.2(i)). For the Ferrers function $\mathsf{P}$ and the associated Legendre function $P$, see §§14.3(i) and 14.21(i). … For collections of integrals of the functions $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$, including integrals with respect to the order, see Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).
##### 7: 15.12 Asymptotic Approximations
15.12.2 $F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),$ $|c|\to\infty$.
15.12.3 $F\left({a,b\atop c+\lambda};z\right)\sim\frac{\Gamma\left(c+\lambda\right)}{% \Gamma\left(c-b+\lambda\right)}\sum_{s=0}^{\infty}q_{s}(z){\left(b\right)_{s}}% \lambda^{-s-b},$
15.12.4 $\left(\frac{{\mathrm{e}}^{t}-1}{t}\right)^{b-1}{\mathrm{e}}^{t(1-c)}\left(1-z+% z{\mathrm{e}}^{-t}\right)^{-a}=\sum_{s=0}^{\infty}q_{s}(z)t^{s}.$
15.12.5 $\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),$
For $I_{\nu}\left(z\right)$ see §10.25(ii). …
##### 8: 16.16 Transformations of Variables
16.16.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,y\right)=(1-y% )^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};\frac{x}{1-y}\right),$
16.16.3 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)=(1-y)^{-% \beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},\beta^{\prime};\gamma;% x,\frac{x}{1-y}\right),$
16.16.4 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,\frac{y}{y% -1}\right),$
16.16.5 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)=(1-y)^{% \alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x+y-xy\right),$
16.16.9 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=(1-% x)^{-\alpha}{F_{2}}\left(\alpha;\gamma-\beta,\beta^{\prime};\gamma,\gamma^{% \prime};\frac{x}{x-1},\frac{y}{1-x}\right),$
##### 9: 30.11 Radial Spheroidal Wave Functions
with $J_{\nu}$, $Y_{\nu}$, ${H^{(1)}_{\nu}}$, and ${H^{(2)}_{\nu}}$ as in §10.2(ii). Then solutions of (30.2.1) with $\mu=m$ and $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$ are given by …Here $a^{-m}_{n,k}(\gamma^{2})$ is defined by (30.8.2) and (30.8.6), and … For fixed $\gamma$, as $z\to\infty$ in the sector $|\operatorname{ph}z|\leq\pi-\delta$ ($<\pi$), … where …
##### 10: 18.3 Definitions
For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … In this chapter, formulas for the Chebyshev polynomials of the second, third, and fourth kinds will not be given as extensively as those of the first kind. … When $j=k=0$ the sum in (18.3.1) is $N+1$. For proofs of these results and for similar properties of the Chebyshev polynomials of the second, third, and fourth kinds see Mason and Handscomb (2003, §4.6). For another version of the discrete orthogonality property of the polynomials $T_{n}\left(x\right)$ see (3.11.9). …