# summation formulas

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##### 2: 1.8 Fourier Series
###### Poisson’s SummationFormula
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: Bibliography V
• A. Verma and V. K. Jain (1983) Certain summation formulae for $q$-series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
• ##### 4: Bibliography B
• 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.
• ##### 5: 1.17 Integral and Series Representations of the Dirac Delta
Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): …
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. However, most of these formulas can be obtained by specialization of formulas for Jacobi polynomials, via (18.7.4)–(18.7.6). In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_{n}\left(x\right)$, $n=0,1,\dots,N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\dots,N+1$, of $T_{N+1}\left(x\right)$: …
The complete cases of $R_{F}$ and $R_{G}$ have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … where both summations extend over the three cyclic permutations of $x,y,z$. Connection formulas for $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ are given in Carlson (1977b, pp. 99, 101, and 123–124). …