Carlitz, Leonard Degenerate Stirling, Bernoulli and Eulerian numbers. (English) Zbl 0404.05004 Util. Math. 15, 51-88 (1979). The author [Arch. Math. 7, 28–33 (1956; Zbl 0070.04003)] defined “degenerate” Bernoulli numbers \(\beta_m(\lambda)\) by means of \[ \frac{x}{(1+\lambda x)^\mu - 1} = \sum_{m=0}^\infty \beta_m(\lambda) \frac{x^m}{m!} \quad (\lambda\mu = 1) \] and proved the following analog of the Staudt-Clausen theorem: \[ \beta_m(a\mid b) = A_m - \sum_{\substack{p-1\mid m \\ p\mid a}} \frac{1}{p} \quad (m \text{ even}), \] where \((a,b) = 1\), \(A_m\) is a rational number whose denominator contains only primes occurring in \(b\), and the summation on the right is over all primes \(p\) such that \(p-1\mid m\) and \(p\mid a\). This result and others proved in the paper suggest that it may be of interest to discuss other sequences of numbers related to \(\beta_m(x)\) and also to obtain additional properties of \(\beta_m(\lambda)\). In view of the close relationship of the Stirling numbers to the Bernoulli numbers we first define \(S(n,k\vert\lambda)\) by \[ \frac1{k!} ((1+\lambda x)^\mu - 1)^k = \sum_{n=0}^k S(n,k\vert\lambda) \frac{x^n}{k!}\quad (\lambda\mu = 1) \] in analogy with the familiar expansion \(\frac1{k!}(e^x - 1)^k = \sum_{n=0}^\infty S(n,k)\frac{x^n}{n!}\), where \(S(n,k)\) is the Stirling number of the second kind. A number of the familiar properties of Stirling number of the second kind are shown to carry over to \(S(n,k\vert\lambda)\). A more interesting result is the multiplication theorem: \[ S(n,j\vert \alpha\beta) = \sum_{k=j}^n \beta^{n-k} S(n,k\vert\alpha) S(k,j\vert\beta), \tag{*} \] where \(\alpha\beta\) are arbitrary. Since \(S(n,k\vert 1) = \delta_{n,k}\) (Kronecker delta), it follows from (*) that \[ \sum_{k=j}^n \mu^{n-k} S(n,k\vert\lambda) S(k,j\vert\mu) = \delta_{n,\ j}, \] where \(\lambda\mu = 1\). This suggests the definition \[ S_1(n,k\vert\lambda)=(-1)^{n-k} \lambda^{n-k} S(n,k\vert\lambda^{-1}), \] so that \[ \sum_{k=j}^n (-1)^{n-k} S(n,k\vert\lambda) S_1(k,j\vert\lambda) = \delta_{n,\ j}, \] analogous to the familiar orthogonality relation \[ \sum_{k=j}^n (-1)^{n-k} S(n,k) S_1(k,j) = \delta_{n,\ j}, \] where \(S_1(k,j)\) denotes a Stirling number of the first kind. The relations \[ S(n,n-k) = \sum_{t=0}^k \binom{k+n}{k-t} \binom{k-n}{k+t} S_1(k+t,t), \] \[ S_1(n,n-k) = \sum_{t=0}^k \binom{k+n}{k-t} \binom{k-n}{k+t} S(k+t,t) \] [for references see “Note on the numbers of Jordan and Ward, Duke Math. J. 38, 783–790 (1971; Zbl 0228.05005)] are shown to carry over to \(S(n,k\vert\lambda)\), \(S_1(n,k\vert\lambda)\). The numbers \(S(n,n - k)\), \(S_1(n,n - k)\) are polynomials of degree \(2k\) in \(n\) and can be written the form \[ S(n,n - k) = \sum_{j=0}^{k-1} S'(k,j) \binom{n}{2k-j} \] \((k\ge 1)\) \[ S(n,n - k) = \sum_{j=0}^{k-1} S'(k,j) \binom{n}{2k-j} \] (for references see above paper). This suggests the analogous definition of \(S'(k,j\vert\lambda)\), \(S'_1(k,j\vert\lambda)\). It has recently been proved by the author [Some number related to the Stirling numbers of the first and second kind, Publ. Fac. Electrotechn. Univ. Belgrade, Ser. Math. Phys. 544–576 (1976), 49–55 (1977; Zbl 0362.05010)] that \[ S(n,n-k)= \sum_{j=1}^k B(k,j) \binom{n+j-1}{2k} \] \((k\ge 1)\) \[ S_1(n,n-k) = \sum_{j=1}^k B_1(k,j) \binom{n+j-1}{2k} \] with \(B_1(k,j) = B(k,k-j+1)\) \((1\le j\le k)\). This result is also shown to carry over: \[ S(n,n-k\vert\lambda) = \sum_{j=1}^k B(k,j\vert\lambda) \binom{n+j-1}{2k} \] \[ S_1(n,n-k\vert\lambda) = \sum_{j=1}^k B_1(k,j\vert\lambda) \binom{n+j-1}{2k} \] with \(B_1(k,j\vert\lambda) = B(k,k - j + 1\vert\lambda)\) \((1\le j\le k)\). The Stirling numbers are related to Bernoulli numbers of higher order [N. E. Nörlund, Vorlesungen über Differenzenrechnung. Berlin: J. Springer (1924; JFM 50.0315.02), Ch. 6] by \[ S(n,n - k) = \binom{n}{k} B_k^{(-n+k)},\quad S_1(n,n - k) = \binom{k-n}{k}B_k^{(n)}. \] The corresponding formulas are \[ S(n,n-k\vert\lambda)= \binom{n}{k} \beta_k^{(-n+k)}(\lambda),\quad S_1(n,n-k\vert\lambda)= \binom{k-n}{k} \beta_k^{(n)}(\lambda), \] where \[ \sum_{m=0}^\infty \beta_m^{(z)}(\lambda) \frac{x^m}{m!} = \left(\frac{x}{(1+\lambda x)^\mu - 1}\right)^z\quad (\lambda\mu = 1) \] The last part of the paper is concerned with the polynomial \(A_{n,k}(\lambda)\) and related functions. The polynomial \(A_{n,k}(\lambda)\) is analogous to the Eulerian number \(A_{n,k}\) [J. Riordan, An introduction to combinatorial analysis. New York etc.: John Wiley (1958; Zbl 0078.00805), p. 39] defined by \[ \frac{1-x}{1 - xe^{(1-x)z}} = 1 + \sum_{n=1}^\infty \frac{z^n}{n!} \sum_{k=1}^n A_{n,k}x^k. \] The corresponding definition of \(A_{n,k}(\lambda)\) is \[ \frac{1-x}{1 - x(1+\lambda(z-xz))^\mu} = 1 + \sum_{n=1}^\infty \frac{z^n}{n!} \sum_{k=1}^n A_{n,k}(\lambda)x^k. \] The symmetry relation \(A_{n,k} = A_{n,n-k+1}\) carries over to \(A_{n,n-k+1}(\lambda) = A_{n,k}(-\lambda)\) \((1\le k\le n)\). Also \(A_{n+1,k}(\lambda)\) satisfies the recurrence \[ A_{n+1,k}(\lambda) = (k - n\lambda) A_{n,k}(\lambda) + (n-k +2 +n\lambda)A_{n,k-1}(\lambda) \] as well as \[ A_{n,k}(\lambda) = n! \sum_{j=0}^k (-1)^j \binom{n+1}{j} \binom{(k-j)\mu}{n} \lambda^n \] and \[ n! \binom{\mu z}{n}^n = \sum_{k=1}^n A_{n,k}(-\lambda) \binom{x+k-1}{n}. \] Reviewer: Leonard Carlitz (Durham, N.C.) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 13 ReviewsCited in 217 Documents MSC: 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers Keywords:degenerate Bernoulli numbers; Stirling numbers; Eulerian numbers Citations:Zbl 0070.04003; Zbl 0228.05005; Zbl 0362.05010; Zbl 0078.00805; JFM 50.0315.02; Zbl 02596078 × Cite Format Result Cite Review PDF