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24 Bernoulli and Euler PolynomialsProperties

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

24.14.1 k=0n(nk)Bk(x)Bn-k(y) =n(x+y-1)Bn-1(x+y)-(n-1)Bn(x+y),
24.14.2 k=0n(nk)BkBn-k =(1-n)Bn-nBn-1.
24.14.3 k=0n(nk)Ek(h)En-k(x) =2(En+1(x+h)-(x+h-1)En(x+h)),
24.14.4 k=0n(nk)EkEn-k =-2n+1En+1(0)
=-2n+2(1-2n+2)Bn+2n+2.
24.14.5 k=0n(nk)Ek(h)Bn-k(x) =2nBn(12(x+h)),
24.14.6 k=0n(nk)2kBkEn-k =2(1-2n-1)Bn-nEn-1.

Let m+n be even with m and n nonzero. Then

24.14.7 j=0mk=0n(mj)(nk)BjBkm+n-j-k+1=(-1)m-1m!n!(m+n)!Bm+n.

§24.14(ii) Higher-Order Recurrence Relations

In the following two identities, valid for n2, the sums are taken over all nonnegative integers j,k, with j+k+=n.

24.14.8 (2n)!(2j)!(2k)!(2)!B2jB2kB2 =(n-1)(2n-1)B2n+n(n-12)B2n-2,
24.14.9 (2n)!(2j)!(2k)!(2)!E2jE2kE2 =12(E2n-E2n+2).

In the next identity, valid for n4, the sum is taken over all positive integers j,k,,m with j+k++m=n.

24.14.10 (2n)!(2j)!(2k)!(2)!(2m)!B2jB2kB2B2m=-(2n+33)B2n-43n2(2n-1)B2n-2.

For (24.14.11) and (24.14.12), see Al-Salam and Carlitz (1959). These identities can be regarded as higher-order recurrences. Let det[ar+s] denote a Hankel (or persymmetric) determinant, that is, an (n+1)×(n+1) determinant with element ar+s in row r and column s for r,s=0,1,,n. Then

24.14.11 det[Br+s] =(-1)n(n+1)/2(k=1nk!)6/(k=12n+1k!),
24.14.12 det[Er+s] =(-1)n(n+1)/2(k=1nk!)2.

See also Sachse (1882).

§24.14(iii) Compendia

For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).