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Divided differences and a general explicit formula for sequences of Mansour–Mulay–Shattuck
Applied Mathematics and Computation 224 (2013) 719–723
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Divided differences and a general explicit formula for sequences of Mansour–Mulay–Shattuck Aimin Xu ⇑, Zhongdi Cen Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China
a r t i c l e
i n f o
Keywords: Divided difference Two-term recurrence Complete Bell polynomial Faà di Bruno’s formula
a b s t r a c t Divided differences is a very important tool, often used both in theory and in practice, for the approximation of functions. In this paper, we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying Mansour–Mulay–Shattuck’s two-term recurrence with arbitrary ðbi ÞiP0 . Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Let N and P denote the non-negative and positive integers, respectively. In [17], Mansour, Mulay and Shattuck studied a class of sequences satisfying a two-term recurrence (also, see [18]). Specifically, suppose ðai ÞiP0 and ðbi ÞiP0 are sequences of complex numbers where the bi are distinct, and let fuðn; kÞgn;kP0 be the array defined by the recurrence
uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to the boundary condition uðn; 0Þ ¼ monly generalizes the following:
Qn1 i¼0
8n; k 2 P;
ð1:1Þ
ðai þ b0 Þ and uð0; kÞ ¼ d0;k for all n; k 2 N. This two-term recurrence com-
Sðn; kÞ ¼ Sðn 1; k 1Þ þ kSðn 1; kÞ; Lðn; kÞ ¼ Lðn 1; k 1Þ þ ðn þ k 1ÞLðn 1; kÞ;
ð1:2Þ
8n; k 2 P;
ð1:3Þ
where Sðn; kÞ are the Stirling numbers of the second kind [20] and Lðn; kÞ are the Lah numbers [19] (also, see [15,27]). Throughout their paper, Mansour and his co-authors devoted themselves to finding the explicit formula for uðn; kÞ and they obtained that if the bi are distinct,
uðn; kÞ ¼
Qn1 k X i¼0 ðbj þ ai Þ ; Qk j¼0 i¼0;–j ðbj bi Þ
8n; k 2 N:
Notice that taking ai ¼ 0 and bi ¼ i in (1.4) for all i yields the Stirling numbers of the second kind
Sðn; kÞ ¼
k k n 1X j ; ð1Þkj k! j¼0 j
while taking ai ¼ bi ¼ i for all i yields the Lah numbers
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A. Xu). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.09.006
ð1:4Þ
720
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
Lðn; kÞ ¼
n! k!
n1
:
k1
See, for example, (6.19) in [13] and (6.3) in [22], respectively. By choosing ai ¼ 0 for all i in (1.4), Mansour et al. rediscovered the explicit expression of the Comtet number [5]. They also applied (1.4) to give a new and unified proof of the explicit formulas for the q-Stirling numbers of Carlitz [1] and the numbers due to Cigler [4]. Other sequences arising as special cases of recurrence (1.1) include the Stirling numbers due to Hsu and Shiue [14] (see also [25]) with ai ¼ ia; bi ¼ r þ ib, and El-Desouky’s multiparameter non-central Stirling numbers [9] with ai ¼ i. For more generalized Stirling numbers, see [7,16,24]. It was pointed out in [17] that if the bi are not all distinct in the recurrence (1.1) above, then it is unclear whether a general formula can be found. Mansour and his co-authors considered though the special case bi ¼ 0 for all i and they showed that uðn; kÞ is given by the ðn kÞth symmetric function on the set fa0 ; a1 ; . . . ; an1 g in this case. Motivated by their interesting work, this short paper is devoted to finding a general explicit formula for uðn; kÞ if the bi are arbitrary complex numbers in the recurrence (1.1). Our work is really a sequel to [17]. ^ ði ¼ 0; 1; . . . ; mÞ is repeated s times, i.e., In the sequence fb0 ; b1 ; . . . ; bk g, we assume that each b i i
0
1
B^ ^ ^ ^ ^ ^ C ðb0 ; b1 ; . . . ; bk Þ ¼ @b 0 ; . . . ; b0 ; b1 ; . . . ; b1 ; . . . ; bm ; . . . ; bm A; |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} s0
s1
sm
^ are distinct. Obviously, s0 þ s1 þ þ sm ¼ k þ 1. For the sake of brevity, we let ½m :¼ f1; 2; . . . ; mg for m P 1, where the b i and let
Am;i ðxÞ :¼
m Y
1
j¼0;–i ðx
C m;p;i ðxÞ :¼
sj
^j Þ b
;
m X ðp 1Þ!sj ; ^ p j¼0;–i ðx b Þ
8i 2 f0g [ ½m; p 2 P:
j
By convention, we set A0;i ¼ 1 and C 0;p;i ¼ 0 for all i and p. Then we can state our main result as follows. Theorem 1.1. Suppose ðai ÞiP0 and ðbi ÞiP0 are sequences of complex numbers. Let fuðn; kÞgn;kP0 be the array defined by the recurrence
uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to the boundary condition uðn; 0Þ ¼
0
Qn1 i¼0
8n; k 2 P;
ðai þ b0 Þ and uð0; kÞ ¼ d0;k for all n; k 2 N. If
1
B^ ^ ^ ^ ^ ^ C ðb0 ; b1 ; . . . ; bk Þ ¼ @b 0 ; . . . ; b0 ; b1 ; . . . ; b1 ; . . . ; bm ; . . . ; bm A; |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} s0
ð1:5Þ
s1
ð1:6Þ
sm
then for all n; k 2 N, we have
uðn; kÞ ¼
si 1 m X X i¼0 j¼0
^Þ Am;i ðb i ^ Þjp 2 ½s 1 j r ^ Y s 1j C m;p;i ðb i i nj bi þ aq1 jq 2 ½n : ðsi 1 jÞ! i
ð1:7Þ
Here Y n ðxp jp 2 ½nÞ :¼ Y n ðx1 ; x2 ; . . . ; xn Þ are the exponential complete Bell polynomials. As an important mathematical tool in enumerative combinatorics, they are defined as follows [6].
X
Y n ðx1 ; x2 ; . . . ; xn Þ :¼
a1 þ2a2 þþnan ¼n
n! xa1 xa2 xann : a1 !ð1!Þa1 a2 !ð2!Þa2 an !ðn!Þan 1 2
It is well known that Y n can be also be expressed as the sum of the exponential partial Bell polynomials Bn;p
Y n ðx1 ; x2 ; . . . ; xn Þ ¼
n X Bn;p ðx1 ; x2 ; . . . ; xnpþ1 Þ: p¼1
For their q-analogs, one can refer to [26]. Another important set of polynomials is the elementary symmetric functions
rp ðxq jq 2 ½nÞ :¼ rp ðx1 ; x2 ; . . . ; xn Þ. They are defined as
rp ðx1 ; x2 ; . . . ; xn Þ :¼
X
xl 1 xl 2 xl p :
16l1
^0 ¼ 0 in (1.7), one has It is worth noticing that if s0 ¼ s1 ¼ ¼ sm ¼ 1, then (1.7) reduces to (1.4). Taking m ¼ 0 and b uðn; kÞ ¼ rn;k ðaq1 jq 2 ½nÞ. In particular, choosing ai ¼ i and ai ¼ ½i for all i gives the Stirling numbers sðn; kÞ of the first kind and their q-analogs [12], respectively.
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
721
2. The proof of the main result In order to obtain (1.7), we first recall two sets of polynomials which were introduced in [17]. The first is called the generalized rising factorial polynomial sequence fqn ðxÞgnP0 defined by
qn ðxÞ ¼
n1 Y
ðx þ ai Þ;
n P 1;
i¼0
with q0 ðxÞ ¼ 1, and the second is the generalized falling polynomial sequence f/n ðxÞgnP0 defined by
un ðxÞ ¼
n1 Y
ðx bi Þ;
n P 1;
i¼0
with /0 ðxÞ ¼ 1. Mansour et al. [17] presented the following important lemma which provides an equivalent way of describing the array uðn; kÞ. Lemma 2.1. Given sequences ðai ÞiP0 and ðbi ÞiP0 in C, the following conditions are equivalent ways to characterize the array fuðn; kÞgn;kP0 :
ðiÞ uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to uð0; kÞ ¼ d0;k
and uðn; 0Þ ¼
X uðn; kÞuk ðxÞ; ðiiÞ qn ðxÞ ¼
n1 Y
n; k 2 P;
ðai þ b0 Þ;
n; k 2 N;
i¼0
n; k 2 N:
kP0
Note that here ðbi ÞiP0 need not all be distinct. This lemma implies that the polynomial qn ðxÞ can be constructed as a linear combination of the basis polynomials uk ðxÞ ðk ¼ 0; 1; . . . ; nÞ and that uðn; kÞ is the coefficient with respect to this basis. Therefore, we may apply the theory of polynomial interpolation (see Chapter 6 in [8]) to derive
uðn; kÞ ¼ Dx ðb0 ; b1 ; . . . ; bk Þqn ;
ð2:1Þ
where Dx ðb0 ; b1 ; . . . ; bk Þ is the divided difference operator acting on functions in the variable x with the nodes b0 ; b1 ; . . . ; bk . According to [8], the divided differences of the function f ðxÞ obey the following recursive formula
Dx ðx0 ; x1 ; . . . ; xn Þf ¼
8 < Dx ðx1 ;x2 ;...;xn Þf Dx ðx0 ;x1 ;...;xn1 Þf ; if xn – x0 ; xn x0 : f ðnÞ ðx0 Þ ;
ð2:2Þ
if xn ¼ x0 ;
n!
where x0 6 x1 6 6 xn . In particular, for distinct points x0 ; x1 ; . . . ; xn , the divided differences of f ðxÞ can be expressed by the explicit formula
Dx ðx0 ; x1 ; . . . ; xn Þf ¼
n X j¼0
f ðxj Þ ; i¼0;–j ðxj xi Þ
Qn
ð2:3Þ
which can be shown by induction. From this explicit expression, one can conclude that Dx ðx0 ; x1 ; . . . ; xn Þf is symmetric with respect to x0 ; x1 ; . . . ; xn . As a consequence, if b0 ; b1 ; . . . ; bk are distinct, then we have
Dx ðb0 ; b1 ; . . . ; bk Þqn ¼
k X j¼0
Qk
qn ðbj Þ
i¼0;–j ðbj
bi Þ
¼
Qn1 k X i¼0 ðbj þ ai Þ : Qk j¼0 i¼0;–j ðbj bi Þ
ð2:4Þ
This yields a new proof of (1.4). For more applications of divided differences in combinatorics, one is referred to [2,3,10,11,21,23,28]. Now, we assume that the sequence fb0 ; b1 ; . . . ; bk g satisfies the condition (1.6) and look for a solution to the recurrence (1.5). First we consider the following lemma. Lemma 2.2. Let f be a function in one variable x. If x0 ; x1 ; . . . ; xpþq are distinct, then
Dx ðx0 ; . . . ; xp ; . . . ; xpþq Þf ¼ Dxp ðxp ; xpþ1 ; . . . ; xpþq ÞDx ðx0 ; x1 ; . . . ; xp Þf : Proof. From (2.3) we have
Dx ðx0 ; x1 ; . . . ; xp Þf ¼ f1 ðxp Þ þ f2 ðxp Þ;
ð2:5Þ
722
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
where
f1 ðxp Þ ¼
p1 X i¼0
ðxi xp Þ
f ðxi Þ Qp1
j¼0;–i ðxi
xj Þ
;
f ðxp Þ f2 ðxp Þ ¼ Qp1 : j¼0 ðxp xj Þ Now, let us evaluate Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 and Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 , respectively. First let gðxÞ ¼ 1=x. Then we have p1 p1 X f ðxi ÞDx ðxp xi ; xpþ1 xi ; . . . ; xpþq xi Þg X ð1Þqþ1 f ðxi Þ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 ¼ ¼ Qp1 Qpþq Qp1 i¼0 i¼0 j¼0;–i ðxi xj Þ j¼p ðxj xi Þ j¼0;–i ðxi xj Þ
¼
p1 X i¼0
f ðxi Þ : j¼0;–i ðxi xj Þ
Qpþq
Applying (2.3) yields
Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 ¼
pþq X i¼p
X f ðxi Þ f ðxi Þ : ¼ Qpþq Qpþq ðx x Þ ðx x Þ i j j j¼0;–i ðxi xj Þ i¼p j¼0 j¼p;–i i pþq
Qp1
Thus, we have
Dxp ðxp ; xpþ1 ; . . . ; xpþq ÞDx ðx0 ; x1 ; . . . ; xp Þf ¼ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 þ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 ¼
p1 X i¼0
X f ðxi Þ f ðxi Þ þ ¼ Dx ðx0 ; . . . ; xp ; . . . ; xpþq Þf ; Qpþq ðx x Þ i j j¼0;–i j¼0;–i ðxi xj Þ i¼p pþq
Qpþq
and this implies (2.5) is true. h Applying Lemma 2.2 we have Lemma 2.3. Let f be a function in one variable x. If xi;j ðj ¼ 0; 1; . . . ; si ; i ¼ 0; 1; . . . ; mÞ are distinct, then
Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf ¼ Dxm;0 ðxm;0 ; . . . ; xm;sm Þ Dx1;0 ðx1;0 ; . . . ; x1;s1 ÞDx0;0 ðx0;0 ; . . . ; x0;s0 Þ Dx ðx0;0 ; . . . ; xm;0 Þf :
ð2:6Þ
Proof. Since Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf is symmetric with respect to xi;j ðj ¼ 0; 1; . . . ; si ; i ¼ 0; 1; . . . ; mÞ, the following holds:
Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf ¼ Dx ðx0;0 ; x1;0 ; . . . ; xm;0 ; x0;1 ; x0;2 ; . . . ; x0;s0 ; . . . ; xm;1 ; xm;2 ; . . . ; xm;sm Þf : Applying Lemma 2.2 on the right hand side of the above yields (2.6). h Let xi :¼ xi;0 and xi;j ! xi for i 2 f0g [ ½m; j 2 ½si . If we take the limit on both sides of (2.6), then we derive
Dx ðx0s0 þ1 ; xs11 þ1 ; . . . ; xsmm þ1 Þf ¼
1 @ sm @ s1 @ s0 Dx ðx0 ; x1 ; . . . ; xm Þf sm s0 !s1 ! sm ! @xm @xs11 @xs00
ð2:7Þ
s þ1
because of (2.2), where xi i means xi is repeated si þ 1 times. Proof of Theorem 1.1. It is trivial that
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q : Dx ðb0 ; b1 ; . . . ; bk Þqn ¼ Dx ðb n m 0 1 According to (2.3) and (2.7), we have
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q ¼ Dx ð b n m 0 1
0 1 m m X 1 @ si 1 @ ^ Y @ sj 1 ^ ^ 1 A qn ðbi Þ ðb bj Þ ^si 1 ^sj 1 i ðs0 1Þ!ðs1 1Þ! ðsm 1Þ! i¼0 @ b j¼0;–i @ b i
¼
m X i¼0
1 @ si 1 ^ ^Þ ; b b q ð ÞA ð i m;i i n ^si 1 ðsi 1Þ! @ b i
where Am;i ðxÞ is defined as in Section 1. Using the Leibniz rule, we have
j
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q ¼ Dx ðb n m 0 1
m X i¼0
si 1 si 1j X si 1 1 ^ @ ^ qðjÞ n ðbi Þ ^si 1j Am;i ðbi Þ: ðsi 1Þ! j¼0 j @b
723
ð2:8Þ
i
^ Þ as We rewrite Am;i ðb i
^ Þ ¼ sgnðA ðb ^ ÞÞ exp Am;i ðb i m;i i
m X
! ^ ^ sj log jbi bj j :
j¼0;–i
Applying Faà di Bruno’s formula which explicitly gives the higher order derivatives of a composite function [6], we obtain the ^si 1j A ðb ^ Þ in terms of the complete Bell polynomials: following closed form for @ si 1j =@ b m;i i i
@ si 1j ^ Þ ¼ A ðb ^ ÞY ^ A ðb i m;i i si 1j C m;p;i ðbi Þjp 2 ½si 1 j ; si 1j m;i ^ @b
ð2:9Þ
i
where the C m;p;i ðxÞ are defined as in Section 1. A straightforward calculation yields
^ ^ qðjÞ n ðbi Þ ¼ j!rnj bi þ aq1 jq 2 ½n ;
ð2:10Þ
which combined with (2.8) and (2.9) gives (1.7). h Acknowledgments We thank the anonymous referees for their careful reading of our manuscript and their many helpful suggestions. We also thank Dr. Huixia Xu for valuable discussions on the revisions of this paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11201430 and 11201382), the Zhejiang Province Natural Science Foundation (Grant No. Y6110310) and the Ningbo Natural Science Foundation (Grant Nos. 2012A610035 and 2012A610036). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000. W.C. Chu, The Cauchy double alternant and divided differences, Electron. J. Linear Algebra 15 (2006) 14–21. W.C. Chu, Divided differences and generalized Taylor series, Forum Math. 20 (2008) 1097–1108. J. Cigler, A new q-analog of Stirling numbers, Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. II 201 (1–10) (1992) 97–109. L. Comtet, Nombres de Stirling generaux et fonctions symetriques, C. R. Acad. Sci. Paris (Ser. A) 275 (1972) 747–750. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974. G. Dattoli, B. Germano, M.R. Martinelli, P.E. Ricci, Touchard like polynomials and generalized Stirling numbers, Appl. Math. Comput. 12 (2012) 6661– 6665. K. David, W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, third ed., American Mathematical Society, 2002. B.S. El-Desouky, Multiparameter non-central Stirling numbers, Fibonacci Quart. 32 (3) (1994) 218–225. M. El-Mikkawy, B. El-Desouky, On a connection between symmetric polynomials, generalized Stirling numbers and the Newton general divided difference interpolation polynomial, Appl. Math. Comput. 138 (2003) 375–385. A.M. Fu, A. Lascoux, q-Identities from Lagrange and Newton interpolation, Adv. Appl. Math. 31 (2003) 527–531. H. Gould, The q-Stirling numbers of the first and second kinds, Duke Math. J. 28 (1961) 281–289. R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, second ed., Addison Wesley, 1989. L.C. Hsu, P.J-S. Shiue, A unified approach to generalized Stirling numbers, Adv. Appl. Math. 20 (1998) 366–384. J. Lindsay, T. Mansour, M. Shattuck, A new combinatorial interpretation of a q-analogue of the Lah numbers, J. Comb. 2 (2) (2011) 245–264. Q.M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728. T. Mansour, S. Mulay, M. Shattuck, A general two-term recurrence and its solution, Eur. J. Comb. 33 (2012) 20–26. T. Mansour, M. Shattuck, A combinatorial approach to a general two-term recurrence, Discrete Appl. Math. 161 (2013) 2084–2094. T.S. Motzkin, Sorting numbers for cylinders and other classification numbers, Proceedings of Symposia in Pure Mathematics, vol. 19, American Mathematical Society, 1971, pp. 167–176. R.P. Stanley, Enumerative Combinatorics, vol. I, Cambridge University Press, 1997. P.P. Tang, A.M. Xu, Generalized Leibniz functional matrices and divided difference form of the Lagrange–Bürmann formula, Linear Algebra Appl. 436 (2012) 618–630. C. Wagner, Generalized Stirling and Lah numbers, Discrete Math. 160 (1996) 199–218. X.H. Wang, A.M. Xu, On matrix and determinant identities for composite functions, J. Comput. Math. 29 (1) (2011) 17–26. A.M. Xu, A Newton interpolation approach to generalized Stirling numbers, J. Appl. Math. 2012 (2012) 1–17 (Article ID 351935). A.M. Xu, Extensions of Spivey’s Bell number formula, Electron. J. Comb. 18 (2012) P77. A.M. Xu, Z.D. Cen, On a q-analogue of Faà di Bruno’s determinant formula, Discrete Math. 311 (2011) 387–392. A.M. Xu, Z.D. Cen, A unified approach to some recurrence sequences via Faà di Bruno’s formula, Comput. Math. Appl. 62 (2011) 253–260. A.M. Xu, C.J. Wang, Some extensions of Faà di Bruno’s formula with divided differences, Comput. Math. Appl. 59 (2010) 2047–2052.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Divided differences and a general explicit formula for sequences of Mansour–Mulay–Shattuck Aimin Xu ⇑, Zhongdi Cen Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China
a r t i c l e
i n f o
Keywords: Divided difference Two-term recurrence Complete Bell polynomial Faà di Bruno’s formula
a b s t r a c t Divided differences is a very important tool, often used both in theory and in practice, for the approximation of functions. In this paper, we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying Mansour–Mulay–Shattuck’s two-term recurrence with arbitrary ðbi ÞiP0 . Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Let N and P denote the non-negative and positive integers, respectively. In [17], Mansour, Mulay and Shattuck studied a class of sequences satisfying a two-term recurrence (also, see [18]). Specifically, suppose ðai ÞiP0 and ðbi ÞiP0 are sequences of complex numbers where the bi are distinct, and let fuðn; kÞgn;kP0 be the array defined by the recurrence
uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to the boundary condition uðn; 0Þ ¼ monly generalizes the following:
Qn1 i¼0
8n; k 2 P;
ð1:1Þ
ðai þ b0 Þ and uð0; kÞ ¼ d0;k for all n; k 2 N. This two-term recurrence com-
Sðn; kÞ ¼ Sðn 1; k 1Þ þ kSðn 1; kÞ; Lðn; kÞ ¼ Lðn 1; k 1Þ þ ðn þ k 1ÞLðn 1; kÞ;
ð1:2Þ
8n; k 2 P;
ð1:3Þ
where Sðn; kÞ are the Stirling numbers of the second kind [20] and Lðn; kÞ are the Lah numbers [19] (also, see [15,27]). Throughout their paper, Mansour and his co-authors devoted themselves to finding the explicit formula for uðn; kÞ and they obtained that if the bi are distinct,
uðn; kÞ ¼
Qn1 k X i¼0 ðbj þ ai Þ ; Qk j¼0 i¼0;–j ðbj bi Þ
8n; k 2 N:
Notice that taking ai ¼ 0 and bi ¼ i in (1.4) for all i yields the Stirling numbers of the second kind
Sðn; kÞ ¼
k k n 1X j ; ð1Þkj k! j¼0 j
while taking ai ¼ bi ¼ i for all i yields the Lah numbers
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A. Xu). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.09.006
ð1:4Þ
720
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
Lðn; kÞ ¼
n! k!
n1
:
k1
See, for example, (6.19) in [13] and (6.3) in [22], respectively. By choosing ai ¼ 0 for all i in (1.4), Mansour et al. rediscovered the explicit expression of the Comtet number [5]. They also applied (1.4) to give a new and unified proof of the explicit formulas for the q-Stirling numbers of Carlitz [1] and the numbers due to Cigler [4]. Other sequences arising as special cases of recurrence (1.1) include the Stirling numbers due to Hsu and Shiue [14] (see also [25]) with ai ¼ ia; bi ¼ r þ ib, and El-Desouky’s multiparameter non-central Stirling numbers [9] with ai ¼ i. For more generalized Stirling numbers, see [7,16,24]. It was pointed out in [17] that if the bi are not all distinct in the recurrence (1.1) above, then it is unclear whether a general formula can be found. Mansour and his co-authors considered though the special case bi ¼ 0 for all i and they showed that uðn; kÞ is given by the ðn kÞth symmetric function on the set fa0 ; a1 ; . . . ; an1 g in this case. Motivated by their interesting work, this short paper is devoted to finding a general explicit formula for uðn; kÞ if the bi are arbitrary complex numbers in the recurrence (1.1). Our work is really a sequel to [17]. ^ ði ¼ 0; 1; . . . ; mÞ is repeated s times, i.e., In the sequence fb0 ; b1 ; . . . ; bk g, we assume that each b i i
0
1
B^ ^ ^ ^ ^ ^ C ðb0 ; b1 ; . . . ; bk Þ ¼ @b 0 ; . . . ; b0 ; b1 ; . . . ; b1 ; . . . ; bm ; . . . ; bm A; |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} s0
s1
sm
^ are distinct. Obviously, s0 þ s1 þ þ sm ¼ k þ 1. For the sake of brevity, we let ½m :¼ f1; 2; . . . ; mg for m P 1, where the b i and let
Am;i ðxÞ :¼
m Y
1
j¼0;–i ðx
C m;p;i ðxÞ :¼
sj
^j Þ b
;
m X ðp 1Þ!sj ; ^ p j¼0;–i ðx b Þ
8i 2 f0g [ ½m; p 2 P:
j
By convention, we set A0;i ¼ 1 and C 0;p;i ¼ 0 for all i and p. Then we can state our main result as follows. Theorem 1.1. Suppose ðai ÞiP0 and ðbi ÞiP0 are sequences of complex numbers. Let fuðn; kÞgn;kP0 be the array defined by the recurrence
uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to the boundary condition uðn; 0Þ ¼
0
Qn1 i¼0
8n; k 2 P;
ðai þ b0 Þ and uð0; kÞ ¼ d0;k for all n; k 2 N. If
1
B^ ^ ^ ^ ^ ^ C ðb0 ; b1 ; . . . ; bk Þ ¼ @b 0 ; . . . ; b0 ; b1 ; . . . ; b1 ; . . . ; bm ; . . . ; bm A; |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} s0
ð1:5Þ
s1
ð1:6Þ
sm
then for all n; k 2 N, we have
uðn; kÞ ¼
si 1 m X X i¼0 j¼0
^Þ Am;i ðb i ^ Þjp 2 ½s 1 j r ^ Y s 1j C m;p;i ðb i i nj bi þ aq1 jq 2 ½n : ðsi 1 jÞ! i
ð1:7Þ
Here Y n ðxp jp 2 ½nÞ :¼ Y n ðx1 ; x2 ; . . . ; xn Þ are the exponential complete Bell polynomials. As an important mathematical tool in enumerative combinatorics, they are defined as follows [6].
X
Y n ðx1 ; x2 ; . . . ; xn Þ :¼
a1 þ2a2 þþnan ¼n
n! xa1 xa2 xann : a1 !ð1!Þa1 a2 !ð2!Þa2 an !ðn!Þan 1 2
It is well known that Y n can be also be expressed as the sum of the exponential partial Bell polynomials Bn;p
Y n ðx1 ; x2 ; . . . ; xn Þ ¼
n X Bn;p ðx1 ; x2 ; . . . ; xnpþ1 Þ: p¼1
For their q-analogs, one can refer to [26]. Another important set of polynomials is the elementary symmetric functions
rp ðxq jq 2 ½nÞ :¼ rp ðx1 ; x2 ; . . . ; xn Þ. They are defined as
rp ðx1 ; x2 ; . . . ; xn Þ :¼
X
xl 1 xl 2 xl p :
16l1
^0 ¼ 0 in (1.7), one has It is worth noticing that if s0 ¼ s1 ¼ ¼ sm ¼ 1, then (1.7) reduces to (1.4). Taking m ¼ 0 and b uðn; kÞ ¼ rn;k ðaq1 jq 2 ½nÞ. In particular, choosing ai ¼ i and ai ¼ ½i for all i gives the Stirling numbers sðn; kÞ of the first kind and their q-analogs [12], respectively.
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
721
2. The proof of the main result In order to obtain (1.7), we first recall two sets of polynomials which were introduced in [17]. The first is called the generalized rising factorial polynomial sequence fqn ðxÞgnP0 defined by
qn ðxÞ ¼
n1 Y
ðx þ ai Þ;
n P 1;
i¼0
with q0 ðxÞ ¼ 1, and the second is the generalized falling polynomial sequence f/n ðxÞgnP0 defined by
un ðxÞ ¼
n1 Y
ðx bi Þ;
n P 1;
i¼0
with /0 ðxÞ ¼ 1. Mansour et al. [17] presented the following important lemma which provides an equivalent way of describing the array uðn; kÞ. Lemma 2.1. Given sequences ðai ÞiP0 and ðbi ÞiP0 in C, the following conditions are equivalent ways to characterize the array fuðn; kÞgn;kP0 :
ðiÞ uðn; kÞ ¼ uðn 1; k 1Þ þ ðan1 þ bk Þuðn 1; kÞ; subject to uð0; kÞ ¼ d0;k
and uðn; 0Þ ¼
X uðn; kÞuk ðxÞ; ðiiÞ qn ðxÞ ¼
n1 Y
n; k 2 P;
ðai þ b0 Þ;
n; k 2 N;
i¼0
n; k 2 N:
kP0
Note that here ðbi ÞiP0 need not all be distinct. This lemma implies that the polynomial qn ðxÞ can be constructed as a linear combination of the basis polynomials uk ðxÞ ðk ¼ 0; 1; . . . ; nÞ and that uðn; kÞ is the coefficient with respect to this basis. Therefore, we may apply the theory of polynomial interpolation (see Chapter 6 in [8]) to derive
uðn; kÞ ¼ Dx ðb0 ; b1 ; . . . ; bk Þqn ;
ð2:1Þ
where Dx ðb0 ; b1 ; . . . ; bk Þ is the divided difference operator acting on functions in the variable x with the nodes b0 ; b1 ; . . . ; bk . According to [8], the divided differences of the function f ðxÞ obey the following recursive formula
Dx ðx0 ; x1 ; . . . ; xn Þf ¼
8 < Dx ðx1 ;x2 ;...;xn Þf Dx ðx0 ;x1 ;...;xn1 Þf ; if xn – x0 ; xn x0 : f ðnÞ ðx0 Þ ;
ð2:2Þ
if xn ¼ x0 ;
n!
where x0 6 x1 6 6 xn . In particular, for distinct points x0 ; x1 ; . . . ; xn , the divided differences of f ðxÞ can be expressed by the explicit formula
Dx ðx0 ; x1 ; . . . ; xn Þf ¼
n X j¼0
f ðxj Þ ; i¼0;–j ðxj xi Þ
Qn
ð2:3Þ
which can be shown by induction. From this explicit expression, one can conclude that Dx ðx0 ; x1 ; . . . ; xn Þf is symmetric with respect to x0 ; x1 ; . . . ; xn . As a consequence, if b0 ; b1 ; . . . ; bk are distinct, then we have
Dx ðb0 ; b1 ; . . . ; bk Þqn ¼
k X j¼0
Qk
qn ðbj Þ
i¼0;–j ðbj
bi Þ
¼
Qn1 k X i¼0 ðbj þ ai Þ : Qk j¼0 i¼0;–j ðbj bi Þ
ð2:4Þ
This yields a new proof of (1.4). For more applications of divided differences in combinatorics, one is referred to [2,3,10,11,21,23,28]. Now, we assume that the sequence fb0 ; b1 ; . . . ; bk g satisfies the condition (1.6) and look for a solution to the recurrence (1.5). First we consider the following lemma. Lemma 2.2. Let f be a function in one variable x. If x0 ; x1 ; . . . ; xpþq are distinct, then
Dx ðx0 ; . . . ; xp ; . . . ; xpþq Þf ¼ Dxp ðxp ; xpþ1 ; . . . ; xpþq ÞDx ðx0 ; x1 ; . . . ; xp Þf : Proof. From (2.3) we have
Dx ðx0 ; x1 ; . . . ; xp Þf ¼ f1 ðxp Þ þ f2 ðxp Þ;
ð2:5Þ
722
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
where
f1 ðxp Þ ¼
p1 X i¼0
ðxi xp Þ
f ðxi Þ Qp1
j¼0;–i ðxi
xj Þ
;
f ðxp Þ f2 ðxp Þ ¼ Qp1 : j¼0 ðxp xj Þ Now, let us evaluate Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 and Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 , respectively. First let gðxÞ ¼ 1=x. Then we have p1 p1 X f ðxi ÞDx ðxp xi ; xpþ1 xi ; . . . ; xpþq xi Þg X ð1Þqþ1 f ðxi Þ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 ¼ ¼ Qp1 Qpþq Qp1 i¼0 i¼0 j¼0;–i ðxi xj Þ j¼p ðxj xi Þ j¼0;–i ðxi xj Þ
¼
p1 X i¼0
f ðxi Þ : j¼0;–i ðxi xj Þ
Qpþq
Applying (2.3) yields
Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 ¼
pþq X i¼p
X f ðxi Þ f ðxi Þ : ¼ Qpþq Qpþq ðx x Þ ðx x Þ i j j j¼0;–i ðxi xj Þ i¼p j¼0 j¼p;–i i pþq
Qp1
Thus, we have
Dxp ðxp ; xpþ1 ; . . . ; xpþq ÞDx ðx0 ; x1 ; . . . ; xp Þf ¼ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf1 þ Dxp ðxp ; xpþ1 ; . . . ; xpþq Þf2 ¼
p1 X i¼0
X f ðxi Þ f ðxi Þ þ ¼ Dx ðx0 ; . . . ; xp ; . . . ; xpþq Þf ; Qpþq ðx x Þ i j j¼0;–i j¼0;–i ðxi xj Þ i¼p pþq
Qpþq
and this implies (2.5) is true. h Applying Lemma 2.2 we have Lemma 2.3. Let f be a function in one variable x. If xi;j ðj ¼ 0; 1; . . . ; si ; i ¼ 0; 1; . . . ; mÞ are distinct, then
Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf ¼ Dxm;0 ðxm;0 ; . . . ; xm;sm Þ Dx1;0 ðx1;0 ; . . . ; x1;s1 ÞDx0;0 ðx0;0 ; . . . ; x0;s0 Þ Dx ðx0;0 ; . . . ; xm;0 Þf :
ð2:6Þ
Proof. Since Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf is symmetric with respect to xi;j ðj ¼ 0; 1; . . . ; si ; i ¼ 0; 1; . . . ; mÞ, the following holds:
Dx ðx0;0 ; . . . ; x0;s0 ; x1;0 ; . . . ; x1;s1 ; . . . ; xm;0 ; . . . ; xm;sm Þf ¼ Dx ðx0;0 ; x1;0 ; . . . ; xm;0 ; x0;1 ; x0;2 ; . . . ; x0;s0 ; . . . ; xm;1 ; xm;2 ; . . . ; xm;sm Þf : Applying Lemma 2.2 on the right hand side of the above yields (2.6). h Let xi :¼ xi;0 and xi;j ! xi for i 2 f0g [ ½m; j 2 ½si . If we take the limit on both sides of (2.6), then we derive
Dx ðx0s0 þ1 ; xs11 þ1 ; . . . ; xsmm þ1 Þf ¼
1 @ sm @ s1 @ s0 Dx ðx0 ; x1 ; . . . ; xm Þf sm s0 !s1 ! sm ! @xm @xs11 @xs00
ð2:7Þ
s þ1
because of (2.2), where xi i means xi is repeated si þ 1 times. Proof of Theorem 1.1. It is trivial that
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q : Dx ðb0 ; b1 ; . . . ; bk Þqn ¼ Dx ðb n m 0 1 According to (2.3) and (2.7), we have
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q ¼ Dx ð b n m 0 1
0 1 m m X 1 @ si 1 @ ^ Y @ sj 1 ^ ^ 1 A qn ðbi Þ ðb bj Þ ^si 1 ^sj 1 i ðs0 1Þ!ðs1 1Þ! ðsm 1Þ! i¼0 @ b j¼0;–i @ b i
¼
m X i¼0
1 @ si 1 ^ ^Þ ; b b q ð ÞA ð i m;i i n ^si 1 ðsi 1Þ! @ b i
where Am;i ðxÞ is defined as in Section 1. Using the Leibniz rule, we have
j
A. Xu, Z. Cen / Applied Mathematics and Computation 224 (2013) 719–723
^ s0 ; b ^ s1 ; . . . ; b ^ sm Þ q ¼ Dx ðb n m 0 1
m X i¼0
si 1 si 1j X si 1 1 ^ @ ^ qðjÞ n ðbi Þ ^si 1j Am;i ðbi Þ: ðsi 1Þ! j¼0 j @b
723
ð2:8Þ
i
^ Þ as We rewrite Am;i ðb i
^ Þ ¼ sgnðA ðb ^ ÞÞ exp Am;i ðb i m;i i
m X
! ^ ^ sj log jbi bj j :
j¼0;–i
Applying Faà di Bruno’s formula which explicitly gives the higher order derivatives of a composite function [6], we obtain the ^si 1j A ðb ^ Þ in terms of the complete Bell polynomials: following closed form for @ si 1j =@ b m;i i i
@ si 1j ^ Þ ¼ A ðb ^ ÞY ^ A ðb i m;i i si 1j C m;p;i ðbi Þjp 2 ½si 1 j ; si 1j m;i ^ @b
ð2:9Þ
i
where the C m;p;i ðxÞ are defined as in Section 1. A straightforward calculation yields
^ ^ qðjÞ n ðbi Þ ¼ j!rnj bi þ aq1 jq 2 ½n ;
ð2:10Þ
which combined with (2.8) and (2.9) gives (1.7). h Acknowledgments We thank the anonymous referees for their careful reading of our manuscript and their many helpful suggestions. We also thank Dr. Huixia Xu for valuable discussions on the revisions of this paper. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11201430 and 11201382), the Zhejiang Province Natural Science Foundation (Grant No. Y6110310) and the Ningbo Natural Science Foundation (Grant Nos. 2012A610035 and 2012A610036). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000. W.C. Chu, The Cauchy double alternant and divided differences, Electron. J. Linear Algebra 15 (2006) 14–21. W.C. Chu, Divided differences and generalized Taylor series, Forum Math. 20 (2008) 1097–1108. J. Cigler, A new q-analog of Stirling numbers, Österreich. Akad. Wiss. Math. Natur. Kl. Sitzungsber. II 201 (1–10) (1992) 97–109. L. Comtet, Nombres de Stirling generaux et fonctions symetriques, C. R. Acad. Sci. Paris (Ser. A) 275 (1972) 747–750. L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974. G. Dattoli, B. Germano, M.R. Martinelli, P.E. Ricci, Touchard like polynomials and generalized Stirling numbers, Appl. Math. Comput. 12 (2012) 6661– 6665. K. David, W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, third ed., American Mathematical Society, 2002. B.S. El-Desouky, Multiparameter non-central Stirling numbers, Fibonacci Quart. 32 (3) (1994) 218–225. M. El-Mikkawy, B. El-Desouky, On a connection between symmetric polynomials, generalized Stirling numbers and the Newton general divided difference interpolation polynomial, Appl. Math. Comput. 138 (2003) 375–385. A.M. Fu, A. Lascoux, q-Identities from Lagrange and Newton interpolation, Adv. Appl. Math. 31 (2003) 527–531. H. Gould, The q-Stirling numbers of the first and second kinds, Duke Math. J. 28 (1961) 281–289. R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, second ed., Addison Wesley, 1989. L.C. Hsu, P.J-S. Shiue, A unified approach to generalized Stirling numbers, Adv. Appl. Math. 20 (1998) 366–384. J. Lindsay, T. Mansour, M. Shattuck, A new combinatorial interpretation of a q-analogue of the Lah numbers, J. Comb. 2 (2) (2011) 245–264. Q.M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728. T. Mansour, S. Mulay, M. Shattuck, A general two-term recurrence and its solution, Eur. J. Comb. 33 (2012) 20–26. T. Mansour, M. Shattuck, A combinatorial approach to a general two-term recurrence, Discrete Appl. Math. 161 (2013) 2084–2094. T.S. Motzkin, Sorting numbers for cylinders and other classification numbers, Proceedings of Symposia in Pure Mathematics, vol. 19, American Mathematical Society, 1971, pp. 167–176. R.P. Stanley, Enumerative Combinatorics, vol. I, Cambridge University Press, 1997. P.P. Tang, A.M. Xu, Generalized Leibniz functional matrices and divided difference form of the Lagrange–Bürmann formula, Linear Algebra Appl. 436 (2012) 618–630. C. Wagner, Generalized Stirling and Lah numbers, Discrete Math. 160 (1996) 199–218. X.H. Wang, A.M. Xu, On matrix and determinant identities for composite functions, J. Comput. Math. 29 (1) (2011) 17–26. A.M. Xu, A Newton interpolation approach to generalized Stirling numbers, J. Appl. Math. 2012 (2012) 1–17 (Article ID 351935). A.M. Xu, Extensions of Spivey’s Bell number formula, Electron. J. Comb. 18 (2012) P77. A.M. Xu, Z.D. Cen, On a q-analogue of Faà di Bruno’s determinant formula, Discrete Math. 311 (2011) 387–392. A.M. Xu, Z.D. Cen, A unified approach to some recurrence sequences via Faà di Bruno’s formula, Comput. Math. Appl. 62 (2011) 253–260. A.M. Xu, C.J. Wang, Some extensions of Faà di Bruno’s formula with divided differences, Comput. Math. Appl. 59 (2010) 2047–2052.