## Congruences for Degenerate Number Sequences

Citations: | 1 - 0 self |

### BibTeX

@MISC{Young_congruencesfor,

author = {Paul Thomas Young},

title = {Congruences for Degenerate Number Sequences},

year = {}

}

### OpenURL

### Abstract

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identification we deduce systems of congruences which are analogues and generalizations of the Kummer congruences for the ordinary Bernoulli numbers. Keywords: Degenerate weighted Stirling numbers; Degenerate Eulerian polynomials; Partial Stirling numbers; Kummer congruences; p-adic integration 1.

### Citations

204 |
Introduction to Cyclotomic Fields
- Washington
- 1982
(Show Context)
Citation Context ...ve integers n, with the convention (xj) 0 = 1. Note that ifs6= 0 then (xj) n = n ( \Gamma1 xj1) n . Proof of Theorem 1.1. Letsdenote the set of all Z p -valued measures on Z p . As is well-known (cf. =-=[19]-=-, [18]), there is a one-to-one correspondences$Z p [[T \Gamma 1]]; (2:2) under which each measure fl 2scorresponds to the formal power series h fl (T ) = Z Z p T x dfl(x): (2:3) The integral in (2.3) ... |

81 |
A course in p-adic analysis
- Robert
- 2000
(Show Context)
Citation Context ...2 Z p by (2.8), (2.12). We now turn to the congruences. It is easily established by induction that if A j B (mod p a Z p [x]) then A p b j B p b (mod p a+b Z p [x]) for any nonnegative integer b (cf. =-=[17]-=-, Prop. VII.3.2.4, p. 407). This principle will be needed for the proof of Theorem 1.2 and of the following proposition, which may be of some independent interest. Proposition 2.1. If p is an odd prim... |

23 |
A unified approach to generalized Stirling numbers
- Hsu, Shiue
- 1998
(Show Context)
Citation Context ...ations ([2], eq. (2.12)) (xj) n = n X k=0 S(n; kj)(xj1) k ; (1:1) where (xj) n is the generalized falling factorial defined in section 2 below. This type of relation has permitted a unified treatment =-=[12]-=- of the various kinds of generalizations of Stirling numbers. The noncentral Stirling numbers and noncentral Lah numbers have been interpreted as higher-order differences of generalized factorials in ... |

21 | Weighted Stirling numbers of the first and second kind–II,” Fibonacci Quart - Carlitz - 1980 |

19 |
A review of the Stirling numbers, their generalizations and statistical applications
- A, Singh
- 1988
(Show Context)
Citation Context ...nd kind R(n; k; r) studied by Carlitz ([3], [4]), and the congruences are valid when the weight r is a p-adic integer. This case includes the noncentral Stirling numbers of the second kind (cf. [14], =-=[6]-=-) and the noncentral Lah numbers [6]. If fi = 1 and r = 0 we have the degenerate Stirling numbers of the second kind and the congruences hold for ff =s2 Z p . If fi = 1 we have the degenerate weighted... |

14 |
Degenerated weighted Stirling numbers
- Howard
- 1985
(Show Context)
Citation Context ... 0 we have the degenerate Stirling numbers of the second kind and the congruences hold for ff =s2 Z p . If fi = 1 we have the degenerate weighted Stirling numbers of the second kind studied by Howard =-=[10]-=-, and the congruences hold when ff =s2 Z p and the weight r is a p-adic integer. For a prime p let T p (n; k; ff; fi; r) be the degenerate number sequence arising from 'h(T ), where h(T ) = T r (T fi ... |

12 |
Non-Central Stirling Numbers and Some Applications
- Koutras
(Show Context)
Citation Context ...e second kind R(n; k; r) studied by Carlitz ([3], [4]), and the congruences are valid when the weight r is a p-adic integer. This case includes the noncentral Stirling numbers of the second kind (cf. =-=[14]-=-, [6]) and the noncentral Lah numbers [6]. If fi = 1 and r = 0 we have the degenerate Stirling numbers of the second kind and the congruences hold for ff =s2 Z p . If fi = 1 we have the degenerate wei... |

7 | A finite difference approach to degenerate Bernoulli and Stirling polynomials, Discrete
- Adelberg
(Show Context)
Citation Context ... factorials in [5]. Combinatorial properties of degenerate Bernoulli and Stirling polynomials have also been effectively studied by considering them as divided differences of binomial coefficients in =-=[1]-=-. The approach of this article will be to instead realize all these generalizations of Stirling numbers, and other combinatorially important numbers, as p-adic integrals of generalized factorials (xj)... |

7 |
On the Differences of the Generalized Factorials at an Arbitrary Point and Their Combinatorial Applications
- Charalambides, Koutras
(Show Context)
Citation Context ... of the various kinds of generalizations of Stirling numbers. The noncentral Stirling numbers and noncentral Lah numbers have been interpreted as higher-order differences of generalized factorials in =-=[5]-=-. Combinatorial properties of degenerate Bernoulli and Stirling polynomials have also been effectively studied by considering them as divided differences of binomial coefficients in [1]. The approach ... |

7 | Some congruences concerning the bell numbers
- Gertsch, Robert
- 1996
(Show Context)
Citation Context ... for all positive integers m and r we have (xj) mp r j x mp r (mod p r+1 Z p [x]): For p = 2, ifs2 2Z 2 then for all positive integers m and r we have (xj) m2 r j x m2 r (mod 2 r Z 2 [x]): Proof. In (=-=[9]-=-, Lemma 1.3) it is shown that for odd primes p, (xj1) p r j (x p \Gamma x) p r\Gamma1 (mod p r Z[x]); (2:13) while for p = 2 we have (xj1) 2 r+1 j (x 2 \Gamma x) 2 r (mod 2 r Z[x]): (2:14) Begin by re... |

7 |
Theory 78
- Bernoulli, Euler, et al.
- 1999
(Show Context)
Citation Context ...d degenerate number sequences arising from the power series 'h(T ), where ' is the linear transformation on Z p [[T \Gamma 1]] defined formally by 'h(T ) = h(T ) \Gamma 1 p X i p =1 h(iT ) (1:9) (cf. =-=[20]-=-, eq. (2.14)). For example, let T p (n; kj) denote the degenerate number sequence arising from the polynomial 'h(T ), where h(T ) = (T \Gamma 1) k . In section 3 we prove congruences implying T p (m; ... |

6 |
Hensel's Lemma and the Divisibility by Primes of Stirling-Like Numbers
- Clarke
(Show Context)
Citation Context ...Z p ) (1:10) for m j n (mod (p \Gamma 1)p a ) ands2 pZ p . In this case the values ats= 0, T p (n; k) = T p (n; kj0), are the "partial" Stirling numbers studied by Lundell [16], Davis [8], a=-=nd Clarke [7]-=-. While many treatments of degenerate number sequences allow for arbitrary complex parameters, the parameter values in many of the important examples are actually rational numbers or integers. For thi... |

6 |
On certain summation problems and generalizations of Eulerian polynomials and numbers
- Hsu, Shiue
- 1999
(Show Context)
Citation Context ...A n (; r; x) by 1 X k=0 (k + rj) n x k = A n (; r; x) (1 \Gamma x) n+1 (4:1) fors6= 0 and x 6= 1. We remark that the polynomial A n (; r; x) was denoted by n!A n (; x; r) in [2] and by A n (x; rj) in =-=[13]-=-. By ([15], Proposition 2.1) we have the identities A n;k (; r) = k X j=0 (\Gamma1) j ` n + 1 j ' (k + r \Gamma jj) n (4:2) and (t + rj) n = n X k=0 A n;k (; r) ` t + n \Gamma k n ' (4:3) where A n (;... |

6 |
A Divisibility Property for Stirling Numbers
- Lundell
- 1978
(Show Context)
Citation Context ...j) j T p (n; kj) (mod p a+1 Z p ) (1:10) for m j n (mod (p \Gamma 1)p a ) ands2 pZ p . In this case the values ats= 0, T p (n; k) = T p (n; kj0), are the "partial" Stirling numbers studied b=-=y Lundell [16]-=-, Davis [8], and Clarke [7]. While many treatments of degenerate number sequences allow for arbitrary complex parameters, the parameter values in many of the important examples are actually rational n... |

4 |
Divisibility by 2 of Stirling-Like Numbers
- Davis
- 1990
(Show Context)
Citation Context ... kj) (mod p a+1 Z p ) (1:10) for m j n (mod (p \Gamma 1)p a ) ands2 pZ p . In this case the values ats= 0, T p (n; k) = T p (n; kj0), are the "partial" Stirling numbers studied by Lundell [1=-=6], Davis [8]-=-, and Clarke [7]. While many treatments of degenerate number sequences allow for arbitrary complex parameters, the parameter values in many of the important examples are actually rational numbers or i... |

4 |
Explicit formulas for degenerate Bernoulli numbers
- Howard
- 1996
(Show Context)
Citation Context ...j y (mod mZ p ) is equivalent to ord p (x \Gamma y)sord p m, and if x and y are rational numbers this congruence for all primes p is equivalent to the definition of congruence x j y (mod m) given in (=-=[11]-=-, x2). The symbolssandswill always represent elements of Q p satisfyings= 1. The generalized falling factorial (xj) n with incrementsis defined by (xj) n = n\Gamma1 Y i=0 (x \Gamma i) (2:1) 4 for posi... |

2 | Eulerian numbers associated with sequences of polynomials, Fibonacci Quart
- Koutras
- 1994
(Show Context)
Citation Context ...x) by 1 X k=0 (k + rj) n x k = A n (; r; x) (1 \Gamma x) n+1 (4:1) fors6= 0 and x 6= 1. We remark that the polynomial A n (; r; x) was denoted by n!A n (; x; r) in [2] and by A n (x; rj) in [13]. By (=-=[15]-=-, Proposition 2.1) we have the identities A n;k (; r) = k X j=0 (\Gamma1) j ` n + 1 j ' (k + r \Gamma jj) n (4:2) and (t + rj) n = n X k=0 A n;k (; r) ` t + n \Gamma k n ' (4:3) where A n (; r; x) = n... |

2 |
On some congruences for the Bell numbers and for the Stirling numbers
- Tsumura
- 1991
(Show Context)
Citation Context ...egers n, with the convention (xj) 0 = 1. Note that ifs6= 0 then (xj) n = n ( \Gamma1 xj1) n . Proof of Theorem 1.1. Letsdenote the set of all Z p -valued measures on Z p . As is well-known (cf. [19], =-=[18]-=-), there is a one-to-one correspondences$Z p [[T \Gamma 1]]; (2:2) under which each measure fl 2scorresponds to the formal power series h fl (T ) = Z Z p T x dfl(x): (2:3) The integral in (2.3) repres... |