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- W2118129087 abstract "In 1738 Euler used the generating functions to study the Bernoulli polynomials. The Bernoulli polynomials were also studied by J.-L. Raabe (1801-1859) [106] and Schlomilch. Raabe found two important formulas for these polynomials. James Stirling (1692–1770), who was born in Scotland, was a contemporary of Euler, who studied in Glasgow and Oxford. Stirling expressed Maclaurins formula in a different form using what is now called Stirling’s numbers of the second kind [53, p. 102]. A.T. Vandermonde (1735–1796) is best known for his determinant and for the Vandermonde theorem for hypergeometric series [137]. Vandermonde also introduced the following notation in 1772 [137]. Let (x)n = x(x − 1)(x − 2) · · · (x − n + 1) be the falling factorial. Stirling numbers of the first kind are the coefficients in the expansion (x)n = ∑n k=0 s(n, k)(x) . The Stirling numbers of the second kind are given by x = ∑n k=0 S(n, k)(x)k. In combinatorics, unsigned Stirling numbers of the first kind |s(n, k)| count the number of permutations of n elements with k disjoint cycles. Tables of the first S(n, k) were given by De Morgan [35, p. 253] and by Grunert [63], [64, p. 279]. These tables were subsequently extended by Cayley [23]." @default.
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- W2118129087 title "q-BERNOULLI AND q-STIRLING NUMBERS, AN UMBRAL APPROACH" @default.
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