Extensions of Spivey's Bell Number Formula
نویسنده
چکیده
We establish an extension of Spivey’s Bell number formula and its associated Bell polynomial extension by using Hsu-Shiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend Gould-Quaintance’s new Bell number formulas.
منابع مشابه
Some extensions of Faà di Bruno's formula with divided differences
Keywords: Bell polynomial Divided difference Faà di Bruno's formula Multicomposite function a b s t r a c t The well-known formula of Faà di Bruno's for higher derivatives of a composite function has played an important role in combinatorics. In this paper we generalize the divided difference form of Faà di Bruno's formula and give an explicit formula for the n-th divided difference of a multic...
متن کاملSpivey's Generalized Recurrence for Bell Numbers
This entry defines the Bell numbers [1] as the cardinality of set partitions for a carrier set of given size, and derives Spivey’s generalized recurrence relation for Bell numbers [2] following his elegant and intuitive combinatorial proof. As the set construction for the combinatorial proof requires construction of three intermediate structures, the main difficulty of the formalization is hand...
متن کاملPoisson, Dobinski, Rota and coherent states
Forty years ago Rota G. C. [1] while proving that the exponential generating function for Bell numbers B n is of the form ∞ n=0 x n n! (B n) = exp(e x − 1) (1) used the linear functional L such that L(X n) = 1, n ≥ 0 (2) Then Bell numbers (see: formula (4) in [1]) are defined by L(X n) = B n , n ≥ 0. Let us notice then that the above formula is exactly the Dobinski formula [2] if L is interpret...
متن کاملImplications of Spivey’s Bell Number Formula
Note that B(n) is the n Bell number while S(n, k) is the Stirling number of the second kind. We have translated Spivey’s notation into Riordan’s familiar notation which we find more preferable in our work [5, 6]. The purpose of this note is to offer a short generating function proof of Equation (1). We discuss this proof in Section 2. We also use Spivey’s formula, in conjunction with the Stirli...
متن کاملA counting formula for certain hierarchical orderings
A hierarchy is a distribution of elements on levels. For a given number of elements a certain hierarchy can be arrangend in a finite number of ways. We provide a counting formula for certain types of hierarchies. Mainly, this formula is a sum over the integer partitions of the number of elements. Furthermore, several basic combinatorical numbers like the Stirling numbers of the second kind, the...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 19 شماره
صفحات -
تاریخ انتشار 2012