# summation by parts

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##### 1: 2.10 Sums and Sequences
###### §2.10(ii) Summation by Parts
The formula for summation by parts is …
##### 2: 1.8 Fourier Series
1.8.16 $\sum_{n=-\infty}^{\infty}e^{-(n+x)^{2}\omega}={\sqrt{\frac{\pi}{\omega}}\*% \left(1+2\sum_{n=1}^{\infty}e^{-n^{2}\pi^{2}/\omega}\cos\left(2n\pi x\right)% \right)},$ $\Re\omega>0$.
##### 3: 25.11 Hurwitz Zeta Function
25.11.28 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}% \frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\mathrm{d}x,$ $\Re s>-(2n+1)$, $s\neq 1$, $\Re a>0$.
##### 4: Bibliography B
• Á. Baricz and T. K. Pogány (2013) Integral representations and summations of the modified Struve function. Acta Math. Hungar. 141 (3), pp. 254–281.
• B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
• B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
• B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
• J. G. Byatt-Smith (2000) The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.
• ##### 5: 18.17 Integrals
18.17.34 $\int_{0}^{\infty}e^{-xz}L^{(\alpha)}_{n}\left(x\right)e^{-x}x^{\alpha}\mathrm{% d}x=\frac{\Gamma\left(\alpha+n+1\right)z^{n}}{n!(z+1)^{\alpha+n+1}},$ $\Re z>-1$.
18.17.38 $\int_{0}^{1}P_{2n}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(\frac{% 1}{2}-\frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}z\right)_{n+1}}},$ $\Re z>0$,
18.17.39 $\int_{0}^{1}P_{2n+1}\left(x\right)x^{z-1}\mathrm{d}x=\frac{(-1)^{n}{\left(1-% \frac{1}{2}z\right)_{n}}}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)_{n+1}}},$ $\Re z>-1$.
18.17.40 $\int_{0}^{\infty}e^{-ax}L^{(\alpha)}_{n}\left(bx\right)x^{z-1}\mathrm{d}x=% \frac{\Gamma\left(z+n\right)}{n!}\*{(a-b)^{n}}a^{-n-z}\*{{}_{2}F_{1}}\left({-n% ,1+\alpha-z\atop 1-n-z};\frac{a}{a-b}\right),$ $\Re a>0$, $\Re z>0$.
18.17.41 $\int_{0}^{\infty}e^{-ax}\mathit{He}_{n}\left(x\right)x^{z-1}\mathrm{d}x=\Gamma% \left(z+n\right)a^{-n-2}{{}_{2}F_{2}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+% \tfrac{1}{2}\atop-\tfrac{1}{2}z-\tfrac{1}{2}n,-\tfrac{1}{2}z-\tfrac{1}{2}n+% \tfrac{1}{2}};-\tfrac{1}{2}a^{2}\right),$ $\Re a>0$. Also, $\Re z>0$, $n$ even; $\Re z>-1$, $n$ odd.
##### 6: 25.5 Integral Representations
25.5.7 $\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)% !}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x% ^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x,$ $\Re s>-(2n+1)$, $n=1,2,3,\dots$.
##### 7: Bibliography Z
• Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
• I. J. Zucker (1979) The summation of series of hyperbolic functions. SIAM J. Math. Anal. 10 (1), pp. 192–206.
• M. I. Žurina and L. N. Karmazina (1964) Tables of the Legendre functions $P_{-\ifrac{1}{2}+i\tau}(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.
• M. I. Žurina and L. N. Karmazina (1965) Tables of the Legendre functions $P_{-1/2+i\tau}(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.
• ##### 8: 35.4 Partitions and Zonal Polynomials
###### Summation
For $\mathbf{T}\in{\boldsymbol{\Omega}}$ and $\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)$, …
##### 9: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.18 $\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}e^{-z}}{\Gamma\left(a\right)}{% \left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum_{k=1}^{% \infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)},$
for $z\to\infty$ in $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$, with $\Re(z-a)\leq 0$ for $P\left(a,z\right)$ and $\Re(z-a)\geq 0$ for $Q\left(a,z\right)$. …
##### 10: Bibliography C
• C. W. Clenshaw, D. W. Lozier, F. W. J. Olver, and P. R. Turner (1986) Generalized exponential and logarithmic functions. Comput. Math. Appl. Part B 12 (5-6), pp. 1091–1101.
• C. W. Clenshaw (1955) A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9 (51), pp. 118–120.