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Shortened recurrence relations for Bernoulli numbers. (English) Zbl 1171.11010

The authors start with two little known results of Saalschütz giving recurrence relations for the Bernoulli numbers B 2n . The first of these results actually is contained in the second one which reads as

k=m n (-1) k+1 (2 2k -2)C(n-m,k-m)B 2k = j=0 m (-1) j 4 n-j (n-j)! 2(n-j)+1C (n) (m,j)

with certain numbers C(n,m) and C (n) (m,j) which may be given explicitly and where 0jmn. The case m=0 gives the first identity.

Then the authors express these numbers C(n,m), C (n) (m,j) in terms of Stirling numbers of both kinds. The also discuss results by P. G. Todorov [J. Math. Anal. Appl. 104, 309–350 (1984; Zbl 0552.10007)] of a similar taste involving Stirling numbers s(k,l) and S(m,j) of the first and second kind respectively.

Finally, using generating functions for the numbers S(n,k) they get for 1mn and k0 the formula

j=m n n+k j-mS(n-j+k+m,k+m)B j =(-1) m k+m j=1 m+1 N(n,m,k,j) k+m-1 j-1,

where N(n,m,k,j)=(n+k)S(m+1,j)S(n+k-1,k+m-j)-mS(m,j)S(n+k,k+m-j). From this they also derive some other formulas.

MSC:
11B68Bernoulli and Euler numbers and polynomials
11B73Bell and Stirling numbers
05A19Combinatorial identities, bijective combinatorics
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