A Newton Interpolation Approach to Generalized Stirling Numbers

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Volume 2012, Article ID 351935,17pages doi:10.1155/2012/351935

Research Article

A Newton Interpolation Approach to Generalized Stirling Numbers

Aimin Xu

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Correspondence should be addressed to Aimin Xu,[email protected] Received 16 November 2011; Revised 8 December 2011; Accepted 20 December 2011 Academic Editor: Carlos J. S. Alves

Copyrightq2012 Aimin Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We employ the generalized factorials to define a Stirling-type pair{sn, k;α, β, r, Sn, k;α, β, r}

which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided diﬀerences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function.

The generalizations of the well-known Dobinski’s formula are further investigated.

1. Introduction

Throughout this paper the following notations will be used. We denote by R the set of real numbers and by C the set of complex numbers. Let α α0, α1, . . . be a vector. If αiiθi0,1, . . ., we denote the vector byθ. We further denote0,1, . . .by 1 and0,0, . . . by 0. Moreover, let us denote the generalized kth falling factorial of x with increment θ byx^k,θ xx−θ· · ·x−kθθ. Particularly, ifθ 1, we writex^k xx−1· · · x−k1.

In mathematics, Stirling numbers of the first and second kind, which are named after James Stirling, arise in a variety of combinatorics problems. They have played important roles in combinatorics. Stirling numbers of the first kind are the coeﬃcients in the expansion xⁿ _n

k0sn, kx^k, and Stirling numbers of the second kind are characterized byxⁿ _n

k0Sn, kx^k.

Over the past few decades, there has been an interest in generalizing and extending the Stirling numbers in mathematics literature. By starting with transformations between generalized factorial involving three arbitrary parameters α, β, and r, Hsu and Shiue 1 introduced the generalized numbersSn, k;α, β, rand unified those generalizations of the Stir- ling numbers due to Riordan 2 , Carlitz 3, 4 , Howard5 , Charalambides-Koutras 6 ,

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Gould-Hopper7 , Tsylova8 , and others. They define a Stirling-type pair{Sn, k;α, β, r, sn, k;α, β, r}by

x^n,αⁿ

k0

S

n, k;α, β, r

x−r^k,β,

x^n,βⁿ

k0

s

n, k;α, β, r

xr^k,α.

1.1

They systematically investigated many basic properties including orthogonality relations, recurrence relations, generating function, and the Dobinski identity for their Stirling numbers. Recently, Comtet9 definessαn, kandSαn, k, the generalized Stirling numbers of the first kind and second kind associate withα0, α1, . . . , α_n−1, by

x−α₀x−α₁· · ·x−α_n−1 n k0

s_αn, kx^k, xⁿⁿ

k0

Sαn, kx−α0x−α1· · ·x−α_k−1.

1.2

El-Desouky10 modified the noncentral Stirling numbers of the first and second kind. He defined the multiparameter noncentral Stirling numbers of the first kind and second kind as follows:

x−α0x−α1· · ·x−α_n−1 n k0

Sn, k;αx^k, 1.3

xⁿⁿ

k0

sn, k;αx−α₀x−α₁· · ·x−α_k−1. 1.4

The recurrence relations, generating functions, and explicit forms for El-Desouky’s Stirling numbers are obtained.

In another direction, Stirling numbers and their generalizations were investigated via diﬀerential operators. Carlitz and Klamkin11 defined the Stirling numbers of the second kind by

xDⁿⁿ

k1

Sn, kx^kD^k, 1.5

whereDis a diﬀerential operatord/dx. Actually, this can be traced back at least to Scherk 12 . In the physical literature, Katriel13 discovered1.5was in connection with the normal ordering expressions in the boson creation operator a^† and annihilation a, satisfying the

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commutation relationa, a^† 1 of the Weyl algebra. Recently, Lang14,15 generalized the stirling numbers of the second kind by the following operator identity:

x^rDⁿⁿ

k1

Sr;n, kx^kD^k, 1.6

where r is a nonnegative integer. He further obtained many properties of these numbers.

More recently, Blasiak et al.16 definedS_r,sn, k, the generalized Stirling numbers of the second kind arising in the solution of the general normal ordering problem for a boson string, as follows

x^rD^sⁿx^nr−s ns ks

Sr,sn, kx^kD^k. 1.7

These numbers were firstly defined by Carlitz17 . More generally, given two sequences of nonnegative integers r r1, r₂, . . . , r_nand s s1, s₂, . . . , s_n, Blasiak18 generalized this formula by

x^rⁿD^sⁿ· · ·x^r²D^s²x^r¹D^s¹x^dⁿ

s1s2···sn

ks1

Sr,sn, kx^kD^k, 1.8

whered_n _n

k1rk−s_k. He gave an explicit formula for the generalized Stirling numbers Sr,sn, k. In19 , a diﬀerent explicit expression for these numbers was presented.

By considering powersV Uⁿ of the noncommuting variablesU,V satisfyingUV V UhV^s, Mansour and Schork20 introduced a new family of generalized Stirling numbers Ss;hn, kas

V Uⁿⁿ

k1

Ss;hn, kV^sn−kkU^k, 1.9

which reduced to the conventional Stirling numbers of second kind and Bell numbers in the cases 0, h 1. As mentioned in 21 , this type of generalized Stirling numbers is not a special case of Howard’s degenerate weight Stirling numbers although they look very similar.

Moreover, for any sequence of real numbersα α0, α₁, . . . , α_n−1and a sequence of nonnegative integers r r0, r1, . . . , r_n−1, by using operational identity22,23 El-Desouky and Caki´c 24 defined a generalized multiparameter noncentral Stirling numbers of the second kindSn, k;α,rby

n−1 j0

x^αⁱδx^−α^j_r_j ⁿ⁻¹

j0

δ−α_j_r_j ^|r|

k0

Sn, k;α,rx^kD^k, 1.10

where|r|r0r1· · ·r_n−1. These numbers reduced to the multiparameter noncentral Stirling numbers of the second kindSn, k;αin1.3if allr_i1.

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As a useful tool, the Newton interpolation with divided diﬀerences was utilized to obtain closed formulas for Dickson-Stirling numbers in the paper 25 provided by the referee. In this paper, we make use of the generalized factorials to define a Stirling-type pair {sn, k;α, β, r, Sn, k;α, β, r}which unifies various Stirling-type numbers investigated by previous authors. By using the Newton interpolation and divided diﬀerences, we obtain the basic properties including recurrence relations, explicit expression, and generating function.

The generalizations of the well-known Dobinski’s formula are further investigated. This paper is organized as follows. In Section 2, we introduce the Newton interpolation and divided diﬀerences. Several important properties of divided diﬀerences are presented. In Section 3, the definitions of a new family of Stirling numbers are given. According to the definitions, the recurrence relation as well as an explicit formula is derived. Moreover, we also investigate the generating function for our generalized Stirling numbers. In views of our results, we rediscover many interesting special cases which are introduced in the above. Finally, in Section 4, the associated generalized Bell numbers and Bell polynomials are presented. Furthermore, a generalized Dobinski’s formula is derived.

2. Divided Differences and Newton Interpolation

For a sequence of pointsα α0, α₁, . . .and allα_i∈RorC, we define

ω_0,αx 1, ω_n,αx ⁿ⁻¹

i0

x−α_i, n1,2, . . . . 2.1

LetNnxbe the Newton interpolating polynomial of degree at mostnthat interpolates a functionfxat the pointα₀, α₁, . . . , α_n; then this polynomial is given as in

Nnx ⁿ

i0Δα0, . . . , αif·ω_i,αx, 2.2

whereΔα0, . . . , α_if is the divided diﬀerence of the ith order of the functionf. As is well known, for the distinct points α0, α1, . . . , αn, the divided diﬀerences of the function f are defined recursively by the following formula:

Δα0f fα0, 2.3 Δα0, . . . , αnf Δα0, . . . , α_n−1f−Δα1, . . . , αnf

α0−αn . 2.4

Divided differences as the coefficients of the Newton interpolating polynomial have played an important role in numerical analysis, especially in interpolation and approximation by polynomials and in spline theory; see 26 for a recent survey. They also have many applications in combinatorics 27, 28 . The divided differences can be expressed by the explicit formula

Δα₀, . . . , α_nfⁿ

i0

fαi _n

j0, /i

αi−αj

. 2.5

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From the above expression it is not diﬃcult to find the divided diﬀerences are symmetric functions of their arguments. In particular, takingα_iαiθθ /0we have

Δα, . . . , αnθf 1

n!θⁿΔⁿ_θfα 1 n!θⁿ

n i0

n

i −1ⁿ⁻ⁱfαiθ, 2.6

whereΔθis the diﬀerence operator with step sizeθ.

Divided diﬀerences can be extended to the cases with repeated points. From the recursive formula2.4, it is clear that ifα0/α1the following holds:

Δα0, α1f fα1−fα0

α₁−α₀ . 2.7

If repetitions are permitted in the arguments and the functionfis smooth enough, then

αlim1→α0Δα₀, α₁f lim

α1→α0

fα1−fα0

α₁−α₀ fα0. 2.8

This gives the definition of first-order divided diﬀerences with repeated points

Δα0, α0ffα0. 2.9

In general, letα0 ≤α1 ≤ · · · ≤αn. Then the divided diﬀerences with repeated points obey the following recursive formula:

Δα₀, . . . , α_nf

⎧⎪

⎪⎨

⎪⎪

⎩

Δα0, . . . , α_n−1f−Δα1, . . . , αnf

α0−αn , ifαn/α0, fⁿα0

n! , if α_nα₀.

2.10

It is evident that divided diﬀerences can be viewed as a discrete analogue of derivatives.

Ifα_n α₀, then all the pointsα₀, α₁, . . . , α_n are the same. In this case,N_nxin2.2is the Taylor polynomial of the function f at the point α0. More generally, if {α0, α1, . . . , αn} {α₀, . . . , α₀

p0

, α₁, . . . , α₁

p1

, . . . , α_m, . . . , α_m

pm

} andp₀p₁· · ·p_m n1 whereα₀, α₁, . . . , α_mare distinct, we define

Ωix ^m

k0, /i

x−α_kpk, S_lix ^m

k0, /i

p_k

α_k−xl, 2.11

withl≥1,0≤i≤m. Recall that the cycle index of symmetric group

Z_nxk Z_nx1, x₂, . . . , x_n

a12a2···nann

1

a₁!1â¹a₂!2â²· · ·a_n!nâⁿxâ₁¹xâ₂²· · ·xâ_nⁿ 2.12

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is one of the essential tools in enumerative combinatorics29 . Using the cycle index of symmetric group, the divided diﬀerences with repeated points can be expressed by the following explicit formula30 see also31 :

Δα₀, . . . , α_nf^m

i0

Ωi

α_i₋₁^pⁱ⁻¹

j0

Z_p_i_−1−j S_!i

α_if^j α_i

j! , 2.13

where

Zpi−1−j S!i

α_i

Zpi−1−j S1i

α_i , S2i

α_i

, . . . , Spi−1−ji α_i

. 2.14

It is well known that the Leibniz formula for higher derivatives is basic and important in calculus. A divided diﬀerence form of this formula given by32 is stated as below. Let hfg. Iffandgare suﬃciently smooth functions, then for arbitrary pointsα₀, α₁, . . . , α_n,

Δα₀, . . . , α_nhⁿ

i0Δα₀, . . . , α_if·Δα_i, . . . , α_ng. 2.15 This formula is called the Steﬀensen formula which is a generalization of the Leibniz formula.

Ifα0 α1 · · ·αn, then the Leibniz formula holds, namely,

hⁿα0

n i0

n

i fⁱα0gⁿ⁻ⁱα0. 2.16

3. Generalized Stirling Numbers

Letα α0, α₁, . . .andβ β0, β₁, . . .be two vectors. We define two kinds of Stirling-type numbers as

ω_n,αx ⁿ

k0

Sn, k;α, β, rω_k,βx−r, 3.1

ω_n,βx ⁿ

k0

sn, k;α, β, rω_k,αxr, 3.2

whereSn, k;α, β, rare called the generalized Stirling numbers of the second kind with the parametersα,β, andr, andsn, k;α, β, rare called the generalized Stirling numbers of the first kind. It is obvious thatSn, k;α, β, r sn, k;β, α,−r. In particular,Sn, k; 0,1,0is the conventional Stirling number of the second kind, andsn, k; 0,1,0is of the first kind.

In this section, making use of divided diﬀerence operator and the Newton interpolation inSection 2, we will investigate orthogonality relations, recurrences, explicit expressions, and generating functions for the generalized Stirling numbers.

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3.1. Basic Properties of the Generalized Stirling Numbers

Firstly, let us consider orthogonality relations of the two kinds of the generalized Stirling numbers. By substituting3.1into3.2and3.2into3.1, one may easily get the following orthogonality relations

n ik

sn, i;α, β, rSi, k;α, β, r δ_n,k, 3.3

n ik

Sn, i;α, β, rsi, k;α, β, r δn,k, 3.4

respectively, where the Kronecker symbolδn,kis defined byδn,k 1 ifn k, andδn,k 0 if n /k. As a consequence, the inverse relations are immediately obtained:

fnⁿ

k0

Sn, k;α, β, rgk⇐⇒gnⁿ

k0

sn, k;α, β, rfk. 3.5

Next, from the definition3.1, one may see thatSn, k;α, β, rcan be viewed as the coeﬃcients of the Newton interpolation of the functionω_n,αat the pointsrβ0, rβ₁, . . . , r βn. Thus, we immediately have the following theorem.

Theorem 3.1. For arbitrary parametersα,β, andr, there holds Sn, k;α, β, r Δ

rβ0, . . . , rβk

ω_n,α. 3.6

In particular, ifβ0, β1, . . . , βnare distinct, we have

Sn, k;α, β, r ^k

i0

_n−1

j0

rβ_i−α_j _k

j0, /i

βi−βj

. 3.7

Ifβ₀β₁· · ·β_n0, then

Sn, k;α,0, r 1 k!

0≤i1<···<in−k≤n−1

r−α_i₁· · ·r−α_i_n−k. 3.8

This theorem gives the explicit expressions for the generalized Stirling numbers. We can similarly getsn, k;α, β, r Δ−rα0, . . . ,−rαkωn,β. By3.6, we can further get the recurrence relations as follows.

Theorem 3.2. For arbitrary parametersα,β, andr, there holds Sn, k;α, β, r Sn−1, k−1;α, β, r

rβ_k−α_n−1

Sn−1, k;α, β, r. 3.9

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In particular, we have

Sn,0;α, β, r ω_n,α rβ0

rβ0−α0

· · ·

rβ0−α_n−1

. 3.10

Proof. According to3.6, we have Sn, k;α, β, r Δ

rβ₀, . . . , rβ_k

ω_n−1,α· −α_n−1. 3.11

By using the Steﬀensen formula for divided diﬀerences and the basic facts

Δx0, . . . , xiωj,αδi,j, 3.12

we have

Sn, k;α, β, r Δ

rβ₀, . . . , rβ_k−1 ω_n−1,α

rβ_k−α_n−1 Δ

rβ₀, . . . , rβ_k

ω_n−1,α. 3.13

This leads to3.9, and the proof is complete.

Finally, let us consider the generating function of the Stirling numbersSn, k;α, β, r denoted byGt;k,α, β, r. Assume thatGt;k,α, β, ris of the form:

Gt;k,α, β, r ^∞

n0

A_nSn, k;α, β, rtⁿ, 3.14

whereA₀, A₁, . . . is a reference sequence. In this way we treat at the same time the case of ordinary coeﬃcients ofGAn1and the case of Taylor coeﬃcientsAn1/n!. LetΦx, t _∞

n0A_nω_n,αxtⁿ. Making use of3.6, we get the following:

Gt;k,α, β, r ^∞

n0

A_nΔ

rβ₀, . . . , rβ_k

ω_n,αtⁿ Δ

rβ₀, . . . , rβ_k

Φ·, t. 3.15

This formula is essential and important for getting the generating function of the generalized Stirling numbers. If we get the analytic expression of Φx, t by choosing special α, the analytic expression ofGt;k,α, β, ris obtained as well.

3.2. Special Cases

Because the parametersα, β, andrare arbitrary, our results contain many interesting special cases. In this part we will investigate these special cases. Some results have been derived and some are new.

Letθ 0, θ, . . .and allβ_i be distinct. According to Theorems3.1and3.2, we have the explicit expressions and the recurrence relations for new generalized Stirling numbers Sn, k;θ, β, r.

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Corollary 3.3. The numbersSn, k;θ, β, rhave the following explicit expression

S

n, k;θ, β, r ^k

i0

_n−1

j0

rβ_i−jθ _k

j0, /i

βi−βj

. 3.16

Corollary 3.4. The numbersSn, k;θ, β, rsatisfy the following recurrence relation

S

n, k;θ, β, r S

n−1, k−1;θ, β, r

rβ_k−n−1θ S

n−1, k;θ, β, r

. 3.17

Forθ 0, θ, . . .andA_n1/n!, ifθ /0 we have Φx, t ^∞

n0

x^n,θtⁿ

n! 1θt^x/θ, 3.18

and ifθ0 we have

Φx, t ^∞

n0

xⁿtⁿ

n! e^xt. 3.19

Thus, by3.15one easily obtain the following theorem.

Theorem 3.5. The sequence{Sn, k;θ, β, r}has the following exponential generating function:

∞ n0

S

n, k;θ, β, rtⁿ n!

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1θt^r/θ^k

i0

1θt^βⁱ^/θ _k

j0, /i

βi−βj

, θ /0,

e^rt k

i0

e^βⁱ^t _k

j0, /i

βi−βj

, θ0.

3.20

Our generalized Stirling numbersSn, k;θ, β, rinclude the Stirling numbers due to Hsu and Shiue1 , EI-Desouky’s multiparameter noncentral Stirling numbers10 , and the so-called Comtet numbers 9 as special cases. Now, let us discuss these special cases as follows.

Example 3.6. Letβ θ : 0, θ, . . .andθ/0. This implies the pointsβ_i are equally spaced with step sizeθ. By Corollaries 3.3and3.4, we immediately get the explicit expression for the generalized Stirling numbers

S

n, k;θ, θ, r 1

k!θ^k k

i0

−1^k−i k

i n−1

j0

rjθ−iθ

3.21

and the recurrence relation S

n1, k;θ, θ, r S

n, k−1;θ, θ, r

rkθ−nθ S

n, k;θ, θ, r

. 3.22

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Forθ /0, the following holds k

i0

1θt^riθ^/θ _k

j0, /i

iθ−jθ 1

k!θ^k1θt^r/θ^k

i0

−1^k−i k

i 1θt^iθ^/θ 1

k!θ^k1θt^r/θ

1θt^θ^/θ−1_k .

3.23

In a similar manner, we can also get the generating function forθ0. Thus, we have

∞ n0

S

n, k;θ, θ, rtⁿ n!

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ 1

k!1θt^r/θ

1θt^θ^/θ−1 θ

_k

, θ /0, 1

k!e^rt

e^θ^t−1 θ

_k

, θ0.

3.24

Here Sn, k;θ, θ, r is equivalent to Sn, k;α, β, r in 1 . As mentioned in 1 , the generalized Stirling numbersSn, k;θ, θ, r contain serval special cases, for example, two kinds of the classical Stirling numbers, the binomial coefficients, the Lah numbers, Carlitz’s two kinds of weighted Stirling numbers4 , Carlitz’s two kinds of degenerate Stirling num- bers3 , Howard’s weighted degenerate Stirling numbers5 , Gould-Hopper’s noncentral Lah numbers 7 , Riordan’s noncentral Stirling numbers 2 , the noncentral C numbers extensively studied by Charalambides and Koutras 6 , Tsylova’s numbers 8 , Todorov’s numbers33 , Nandi and Dutta’s associated Lah numbers34 , and ther-Stirling numbers of the first kind fully developed by Broder35 . Hsu and Shiue obtained the recurrence relation for the generalized Stirling numbers Sn, k;θ, θ, r, and they also found the generating function by solving a difference-differential equation. However, the formula3.21was new and not given by1 . Obviously, in the present paper we follow a very different approach to rediscover the recurrence relation and the generating function. In the caser0, one may also refer to36 .

It is remarkable that Mansour and Schork20 recently consideredUV −V U hV^s to generalize the commutation relation UV −V U 1. They defined generalized Stirling numbersSs;hn, kby1.9. The explicit expressions of these generalized Stirling numbers are given by20 see also21 , and they are very closely related to the numbers considered by Lang14 . In 21 , the authors exploited many properties of these generalized Stirling numbers. It is interesting that observing our generalized Stirling numbersSn, k;θ, β, rby θ−sh, βkkh1−sandr 0, we find that the Stirling numbers due to Mansour and Schork are actually a special case of ours and Hsu-Shiue’s, and they are equivalent to the numbers due to36 . Thus, by3.20we get the exponential generating function of the generalized Stirling numbers due to Mansour and Schork:

1 k!

1−hst^s−1/s−1 h1−s

k

^∞

n0

Ss;hn, ktⁿ

n!. 3.25

In21 , the authors gave the generating function fork1.

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Example 3.7. Let θ 1, r 0, and βi be arbitrary but distinct, and one can get the multiparameter noncentral Stirling numbers of the first kind introduced by El-Desouky 10 . Here we denote the numbers by Sn, k; 1,β,0. Using Corollaries 3.3 and 3.4 and Theorem 3.5, we rediscover the explicit expression, recurrence relation, and the generating function, namely,

S

n, k; 1,β,0 ^k

i0

_n−1

j0

βi−j _k

j0, /i

βi−βj

, S

n1, k; 1,β,0 S

n, k−1; 1,β,0

βk−n S

n, k; 1,β,0 , ∞

n0

S

n, k; 1,β,0tⁿ n! ^k

i0

1t^βⁱ _k

j0, /i

β_i−β_j.

3.26

Example 3.8. Let us consider the caseθ0. In this case, there holds

xⁿⁿ

k0

Sn, k; 0,β, rωk,βx−r, 3.27

which is equivalent to

xrⁿⁿ

k0

Sn, k; 0,β, rωk,βx. 3.28

Especially, forr 0 the Comtet numbers9 see also 10 are defined associated with the sequenceβby

xⁿⁿ

k0

S_βn, k

x−β₀ x−β₁

· · ·

x−β_k−1

. 3.29

This impliesSn, k; 0,β,0 S_βn, k. Thus, it is not diﬃcult to obtain

Sn, k; 0,β,0 ^k

i0

βⁿ_i _k

j0, /i

β_i−β_j,

Sn1, k; 0,β,0 Sn, k−1; 0,β,0 βkSn, k; 0,β,0,

3.30

and the exponential generating function ∞

n0

Sn, k; 0,β,0tⁿ n! ^k

i0

e^βⁱ^t _k

j0, /i

βi−βj

. 3.31

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It is worth noting that the Comtet numbers Sn, k; 0,β,0can be rewritten as an alternate form

Sn, k; 0,β,0

0≤i1≤···≤in≤k

β_i₁β_i₂· · ·β_i_n, 3.32

which is really the complete symmetric function ofnth order with respect to the variablesβ₀, β₁,. . .,β_k. By3.15, they have the ordinary generating function:

∞ n0

Sn, k; 0,β,0tⁿ Δ

β0, . . . , βk

1

1−·t t^k _k

i0

1−β_it. 3.33

Moreover, if we letr −a,β 1, then we get the noncentral Stirling numbers of the second kind defined by Koutras37 see also24 . For more details one refers to37 .

What has been discussed above in this subsection is relevant to the generalized Stirling numbers with equidistance parametersα_i. However, we are also interested in the other cases.

In a recent year, many authors14,16,18,19,24 were devoted to the generalized Stirling numbers by diﬀerential operator. We here rediscover these generalized Stirling numbers by the Newton interpolation.

Example 3.9. Our generalized Stirling numbersSn, k;α, β, ralso contain the numbers due to Blasiak18 as a special case. Here we let r r1, r2, . . . , rm, s s1, s2, . . . , smand letd0 0 anddm_m

i1ri−siform≥1. Moreover, we let

d−d0, d₀−1, . . . , d₀−s₁1, d₁, d₁−1, . . . , d₁−s₂1, . . . , d_m−1, d_m−1−1, . . . , d_m−1−sm1,

s 0,1, . . . , s1s2· · ·sm,

3.34

wheres1s2· · ·sm n. By using3.7we immediately have the explicit expression of Sn, k;d, s,0as follows

S

n, k;d, s,0 1

k!

k j0

k

j −1^k−j^m

i1

d_i−1j_s_i

, 3.35

which is in accordance with the generalized Stirling numbersS_r,sm, kintroduced by Blasiak 18 . Blasiak got this explicit formula by using the operator x^r^mD^s^m· · ·x^r²D^s²x^r¹D^s¹ to act one^x. His proof is very diﬀerent from ours. Recently, El-Desouky et al. 19 found a new expression by successive application of Leibniz formula. The special case r r, r, . . . , rand s s, s, . . . , sis investigated by Blasiak et al.16 , and they gave us Lang’s result14 as a special case for s 1,1, . . . ,1.

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Example 3.10. By operating with1.10one^xand using Cauchy rule of multiplication of series, El-Desouky and Caki´c24 obtain the explicit formula

Sn, k;α,r 1 k!

k i0

k

i −1^k−iⁿ⁻¹

j0

k−αj

_r_j

. 3.36

In fact, letα {α 0, . . . , α0 r0

, α 1, . . . , α1 r1

, . . . , α_n−1, . . . , α _n−1

rn−1

}. It is not diﬃcult to findSn, k;α,r Sr0r₁· · ·r_n−1, k;α, 1,0holds. In particular, settingn 2,α₀ 0,α₁ 1,r₀ l, and r1 m−linSr0r1· · ·r_n−1, k;α, 1,0, we get the explicit expression of the number of partitions ofM {x1, x₂, . . . , x_m}intonnonempty parts such that the distance of any two members in the same part diﬀers fromldenoted byT_lm, k; see38 .

4. Generalized Bell Polynomials and Dobinski-Type Formulas

Recall that the Bell numbersB_n and the exponential polynomialsB_nxare defined, respectively, by the sums

B_nⁿ

k0

Sn, k, B_nx ⁿ

k0

Sn, kx^k. 4.1

The Bell polynomialsBnxhave the generating function ∞

n0

Bnxtⁿ

n! e^xe^t⁻¹. 4.2

They also satisfy the following remarkable Dobinski-type formula

B_nx e^−x ∞ i0

iⁿ

i!xⁱ, 4.3

which reduces to the Dobinski formula whenx1. It is worth noting thatB_nxis represent- ed as an infinite series ini.

As we know, the Dobinski-type formulas have been the subject of much combinatorial interest. Thus, it is worth looking for a general Dobinski-type formula.

In this section, we define a generalized Bell polynomials by

B_n;α,θ,rx ⁿ

k0

S

n, k;α, θ, r

x^k, 4.4

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whereθ 0, θ, . . .andθ/0. Naturally, one get an extended definition of generalized Bell numbers as follow:

B_n;α,θ,rⁿ

k0

S

n, k;α, θ, r

. 4.5

Note thatB_n;0,1,0x BnxandB_n;0,1,0Bn. We can make use of3.6to obtain the following Dobinski-type formula.

Theorem 4.1. Forθ 0, θ, . . .and arbitraryα, r, we have the Dobinski-type formula

B_n;α,θ,rx e^−x/θ ∞

i0

ω_n,αriθ i!

x θ

i

, 4.6

whereω_n,αis defined by2.1.

Proof. By4.4we have ∞

n0

B_n;α,θ,rxtⁿ n! ^∞

n0

tⁿ n!

n k0

S

n, k;α, θ, r

x^k^∞

k0

x^k

n≥k

S

n, k;α, θ, rtⁿ

n!. 4.7

Replacingβbyθin3.6yields ∞

n0

B_n;α,θ,rxtⁿ n! ^∞

k0

x^k

n≥k

Δ

r, rθ, . . . , rkθ ω_n,αtⁿ

n!

^∞

n0

tⁿ n!

∞ k0

1 k!

x θ

k k

i0

−1^k−i k

i ω_n,α riθ

.

4.8

By equating the coeﬃcient oftⁿ/n! within the first and last expressions, we arrive at

B_n;α,θ,rx ^∞

k0

1 k!

x θ

k k

i0

−1^k−i k

i ω_n,α riθ

. 4.9

Using the Cauchy product rule gives

B_n;α,θ,rx ^∞

j0

−1^j j!

x θ

j∞ i0

ω_n,αriθ i!

x θ

i

. 4.10

This implies4.6is true and completes the proof.

Lettingx1 we directly obtain the generalized Dobinski formula.

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Corollary 4.2. Forθ 0, θ, . . .and arbitraryα, r, we have

B_n;α,θ,re^−1/θ ∞

i0

ω_n,αriθ i!

1 θ

i

, 4.11

whereω_n,αis defined by2.1.

It is clear that4.3is a special case of4.6withθ1,α0, r 0.

Letφ_α,θt _∞

i0iθ−α0· · ·iθ−α_n−1tⁱ/i!. It is worth noting that the formula4.6 can be used to obtain a closed sum formula for this type of infinite series. As mentioned in 1 , such a type of series cannot be summed by using the hypergeometric series method. Let tx/θ, r 0; then according to4.6we have

φ_α,θt B_n;α,θ,0

θt e^te^t

n k0

S

n, k;α, θ,0

θ^kt^k. 4.12

Example 4.3. Lettingαθ 0, θ, . . .we immediately obtain the Dobinski-type formula due to Hsu and Shiue1 as follows:

B_n;θ,θ,rx e^−x/θ ∞

i0

θⁿriθ/θⁿ i!

x θ

i

. 4.13

Example 4.4. Lettingαd, θ 1, ns₁s₂· · ·s_mwe have the following Dobinski-type formula due to Blasiak18 :

B_n;_d,1,0 x e^−x ∞ is1

m j1

d_j−1i_s_jxⁱ

i!. 4.14

Acknowledgments

The author thanks the anonymous referees for their valuable suggestions and comments.

This work was supported by the Zhejiang Province Natural Science FoundationGrant nos.

Y6110310 and Y6100021and the Ningbo Natural Science Foundation.

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