# explicit formulas

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##### 1: 24.6 Explicit Formulas
###### §24.6 ExplicitFormulas
24.6.6 $E_{2n}=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{% \left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j}(k-2j)^{2n}.$
##### 2: 10.49 Explicit Formulas
###### §10.49(i) Unmodified Functions
10.49.7 ${\mathsf{h}^{(2)}_{n}}\left(z\right)=e^{-iz}\sum_{k=0}^{n}(-i)^{k-n-1}\frac{a_% {k}(n+\frac{1}{2})}{z^{k+1}}.$
##### 3: 27.13 Functions
Explicit formulas for $r_{k}\left(n\right)$ have been obtained by similar methods for $k=6,8,10$, and $12$, but they are more complicated. …Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities. …
##### 4: 3.8 Nonlinear Equations
###### §3.8(iv) Zeros of Polynomials
Explicit formulas for the zeros are available if $n\leq 4$; see §§1.11(iii) and 4.43. No explicit general formulas exist when $n\geq 5$. …
##### 5: Bibliography H
• K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.
• F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.
##### 7: Bibliography N
• G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
• ##### 8: 29.15 Fourier Series and Chebyshev Series
29.15.50 $\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=\operatorname{cn}\left(z,k\right)% \operatorname{dn}\left(z,k\right)\sum_{p=0}^{n}D_{2p+2}U_{2p+1}\left(% \operatorname{sn}\left(z,k\right)\right).$
For explicit formulas for Lamé polynomials of low degree, see Arscott (1964b, p. 205).
##### 9: 10.74 Methods of Computation
In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …
##### 10: Bibliography T
• P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.