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一般化されたスターリング数について

On generalized Stirling numbers

中 村 宗 敬

Munetaka NAKAMURA

Abstract

In this note the generalized Stirling numbers of second kind are used for the representations of limit function of the generalized Bernstein polynomials.

1. Introduction

 The normal Stirling numbers of first kind, which is denoted by c（n,k） in this note, is characterized as the following equalities for n 1 ;

x（x＋1）（x＋2）…（x＋n−1）＝

Σ

c（n,k）xk , while on the other hand, the Stirling number of second kind s（n,k） is defined by .

xn＝

_Σ

S（n,k）x（x−1）…（x−k＋1）

for n 1. It is known that he numbers c（n,k） and S（n,k） satisfy the following recursive relations c（n＋1,k）=nc（n,k−1）＋c（n,k） n 1, k 1

S（n＋1,k）=S（n,k−1）＋kS（n,k） n 1, k 1

with an appropriate initial conditions. In ［5］ we used the above numbers and used them to represent Bernstein polynomials and obtained their elaborate asymptotic behavior. See also ［1］.

   The above Stirling numbers are generalized to cq（n,k） and S（q n,k） by the use of the q -integer ［n］＝1＋q＋q2

＋…＋qn−1（0＜q＜1）. See ［3］. They are defined by the relations c（q n,k）＝qn−1c（q n−1,k−1）＋［n−1］c（q n−1,k）, n 2, 1 k n , （3.2） S（q n,k）＝qk−1S（q n−1,k−1）＋［k］S（q n−1,k）, n 2, 1 k n , （3.3）

with the conditions c（1,1）＝_q S（1,1）＝1 and q c（q n,k）＝S（q n,k）＝0 if k 0 or k＞n . Then the Bernstein polynomial Bq,nƒ（x）＝

_Σ

C（q n,k）ƒ（［k］/［n］）x（1−k x）qn−k for a continuous function ƒ on ［0,1］ （C（_q n,k）＝       , ［n］!＝

_Π

［k］ ） is represented through the generalized Sirling number of second kind as follows （［4］） ;

Bq,nƒ（r x）＝  

Σ

S（q r,s）［n］（s） xs  for ƒ（r x）＝xr , ─ 9 ─ k＝0 n k＝1 n k＝1 n k＝1 n ［n］! ［k］!［n−k］! 1 ［n］r

where ［n］（s） ＝

Π

［n−j］ . These forms correspond to those of the ordinary Bernstein polynomials in ［1］. For the outline of properties of the generalized Bernstein polynomials, see ［6］.

 Hereafter in this note, we use this representation theorem to obtain the concrete form of the limit function B∞,nƒ（x）＝ lim Bq,nƒ（r x）. Note that q is fixed while letting n → ∞.

  For the convergence of these generalized Bernstein polynomials, in ［2］, Il inskii and Ostrovska obtained, for example, if 0＜q＜1,

B∞,nƒ（x）＝

Σ

ƒ（x−qk）p∞,k（x） （0 x＜1）, B∞,nƒ（1）＝ƒ（1）

where p∞,k（x）＝      

Π

（1−qsx）. We should remark that their view and representation seems to be probabilistic, while our calculation is rather combinatorial.

2. Representation of the limit functions by Stirling numbers

 Firstly we again quote the following theorem to make our arguments clear.

THEOREM 1. The following equality holds for ƒ（r x）＝xr where r is a positive integer, Bq,nƒ（r x）＝   

Σ

S（q r,s）［n］（s）xs .

THEOREM 2. Let 0＜q＜1 . Then, for a positive integer r ,

B_q,∞ƒ（_r x）＝

Σ

S（_q r,s）（1−q）r−sxs .

Proof. Since ［n］＝1＋q＋q2_＋…＋qn−1 _{→    , as （ordinary） integer n goes to infinity,}

we easily see that    →     ＝（1＋q）r−s , which completes the proof in view of Theorem1. REMARK 1. Formally putting q＝1 , we see that the theorem coincides with the ordinary case. For （1−q）r−s vanishes unless r＝s and this implies B_q,∞ƒ（x）＝S（q r,r） xr ＝ xr in the theorem.

REMARK 2. Theorem 1 and Il inskii-Ostrovska theorem imply that

Σ

S（q r,s）（1−q）r−s ＝1 （This can be seen directly away from these theorems）. In fact, firstly the definition S（1,1）＝1 coincides with the equality. q Furthermore, by the definition, we can recursively compute the Stirling numbers as follows.

  S（2,1）＝1 , q S（2,2）＝q q,   S（3,1）＝1 ,q S（3,2）＝q q2＋2q, S（3,3）＝q q3 ,,   S（4,1）＝1 ,q S（4,2）＝q q3＋3q2＋3q , S（4,3）＝q q5＋2q4＋3q3 , S（4,4）＝q q6 , which go to   S（2,1）q （１−q）＋S（2,2）＝1−q q＋q＝1 , 平成 21 年（2009年）度 山 梨 大 学 教 育 人 間 科 学 部 紀 要 第 11 巻 ─ 10 ─ j＝0 s−1 n→∞ k＝0 ∞ s＝1 r s＝1 r s＝1 r s＝0 ∞ xk （1−q）［k k］! 1 ［n］r ［n］（s） ［n］r 1 １−q （1−q）−s （1−q）−r

  S（3,1）_q （１−q）2＋S（3,2）_q （１−q）＋S（3,3）＝（１−_q q）2＋（q2＋2q）（１−q）＋q3＝1 ,   S（4,1）q （１−q）3＋S（4,2）q （１−q）2＋S（4,3）q （１−q）S（4,4）q

      ＝（１−q）3＋（q3＋3q2＋3q）（１−q）2＋（q5＋2q4＋3q3）（１−q）＋q6＝1.

References

［1］ U.Abel, The complete asymptotic expansion for the Meyer-Konig and Zeller operators, J. Mathematical Analysis and applications 208,109-119, 1997.

［2］ A.Il inskii, S.Ostrovska, Convergence of the generalized Bernstein polynomials, J. Approx. Theory 116,100-112,2002.

［3］ H.W.Gold, The q−Stirling numbers of first and second kind, Duke Math J. 28, 281-289, 1961

［4］ M. Nakamura, On the combinatorial representations of Bernstein operators, Bulletin of the Faculty of Education & Human Science, 9, 7-13, 2008.

［5］ G.M. Philips, Interpolation and approximation by polynomials, Canad. Math. Soc. 2003.

一般化されたスターリング数について （中村）