A Symbolic Operator Approach to Power Series Transformation-Expansion Formulas

Transcription

1 Journal of Integer Sequences, Vol , Article A Symbolic Operator Approach to Power Series Transformation-Expansion Formulas Tian-Xiao He 1 Department of Mathematics and Computer Science Illinois Wesleyan University Bloomington, IL USA the@iwu.edu Abstract In this paper we discuss a ind of symbolic operator method by maing use of the defined Sheffer-type polynomial sequences and their generalizations, which can be used to construct many power series transformation and expansion formulas. The convergence of the expansions are also discussed. 1 Introduction The closed form representation of series has been studied extensively. See, for examples, Comtet [1], Ch. Jordan [11], Egorechev [2], Roman-Rota [15], Sofo [17], Wilf [18], Petovše- Wilf-Zeilberger s boo [13], and the author s recent wor with Hsu, Shiue, and Toney [4]. This paper is a sequel to the wor [4] and the paper with Hsu and Shiue [5], in which the main results are special cases of Theorem 2 shown below. The obect of this paper is to mae use of the following defined generalized Sheffer-type polynomial sequences and the classical operators difference, E shift, and D derivative to construct a method for the summation of power series expansions that appears to have a certain wide scope of applications. As an important tool in the calculus of finite differences and in combinatorial analysis, the operators E,, D are defined by the following relations. 1 The research of this author was partially supported by Artistic and Scholarly Development ASD Grant and sabbatical leave of the IWU. 1

2 Eft ft + 1, ft ft + 1 ft, Dft d dt ft. Powers of these operators are defined in the usual way. In particular for any real numbers x, one may define E x ft ft + x. Also, the number 1 may be used as an identity operator, viz. 1ft ft. Then it is easy to verify that these operators satisfy the formal relations cf. [11] E 1 + e D, E 1 e D 1, D log1 +. Note that E f0 [ E ft ] t0 f, so that xe f0 fx. This means that xe with x as a parameter may be used to generate a general term of the series fx. Definition 1. Let At, Bt, and gt be any formal power series over the real number field R or complex number field C with A0 1, B0 1, g0 0, and g 0 0. Then the polynomials p n x n 0, 1, 2, defined by the generating function GF AtBxgt n 0 p n xt n 1 are called generalized Sheffer-type polynomials associated with At, Bt, gt. Accordingly, p n D with D d/dt is called Sheffer-type differential operator of degree n associated with At, Bt, and gt. In particular, p 0 D I is the identity operator due to p 0 x 1. In Definition 1, if Bt e t, then the defined {p n x} is a classical Sheffer-type polynomial sequence associated with At, gt At, expt, gt. As examples, classical Sheffer-type polynomials include Bernoulli polynomials, Euler polynomials, and Laguerre polynomials generalized by At,gt t/e t 1,t, 2/e t +1,t, and 1 t p,t/t 1 p > 0, respectively. We call the infinite matrix [d n, ] n, 0 with real entries or complex entries a generalized Riordan matrix the originally defined Riordan matrices need g 0 1 if its th column satisfies that is, d n, t n Atgt ; 2 n 0 d n, [t n ]Atgt, the nth term of the expansion of Atgt. The Riordan matrix is denoted by At,gt or [d n, ] described in 2. Then the generalized Sheffer-type polynomial sequence associated with At,Bt,gt is the result of the following matrix multiplication 2

3 [d n, ] 1 b 1 x b 2 x 2. b n x n If [ d n, [t n ]Atgt ] and [ c n, [t n ]Ctft ] are two Riordan matrices, and {p n x} and {q n x} are two corresponding generalized Sheffer-type polynomial sequences associated with At,Bt,gt and Ct,Bt,ft, respectively, then we can define a umbral composition cf. its special case of Bt e t is given in [14] and [15] between {p n x} and {q n x}, denoted by {p n x}#{q n x}. The resulting sequence is the generalized Sheffertype polynomial sequence associated with AtCgt, Bt,fgt. Clearly, the sequence {p n x}#{q n x} is the result of the following matrix multiplication [d n, ][c n, ].. 1 b 1 x b 2 x 2. b n x n A power series Bt b t is said to be regular if b 0 for all 1. Under the composition operator #, it can be proved that all generalized Sheffer-type polynomial sequences associated with a regular Bt form a group, called the generalized Sheffer group associated with Bt. Its verification is analogous to the classical Sheffer group associated with e t established in [14] see also in [6]. In [6], the author established the isomorphism between the classic Sheffer group and the Riordan group based on the following biective mapping: θ : [d n, ] {p n x} or θ : At,gt {p n x}, i.e., θ[d n, ] n 0 :.. d n, x /! [d n, ] n 0 X, 3 for fixed n, where X 1,x,x 2 /2!,... T, or equivalently, θat,gt : [t n ]Ate xgt 4 It is clear that 1, t, the identity Riordan array, maps to the identity Sheffer-type polynomial sequence {p n x x n /} n 0. From the definitions 2 we immediately now that p n x [t n ]Ate xgt if and only if d n, [t n ]At gt. 5 3

4 Therefore, the Riordan matrices from the sequences shown in the On-Line Encyclopedia of Integer Sequences OLEIS also present the coefficients of the corresponding classic Sheffertype polynomials. For example, sequence A , 1, 1, 3, 2, 1, 13, 9, 3, 1, 73, 52, 18, 4, 1,... presents both Riordan array e x/1 x,x and the corresponding Sheffer-type polynomial sequence: p 0 x 1 p 1 x 1 + x p 2 x 3 + 2x + x 2 /2! p 3 x x + 3x 2 /2! + x 3 /3!... The row sums of Riordan array e x/1 x,x, i.e., the polynomial values at x 1, are A052844, while diagonal sums of the array are A cf. [16]. Other examples, including, A000262, A084358, A133289, etc., can be also found in [16]. Suppose that Φt is an analytic function of t or a formal power series in t, say Φt c t, c [t ]Φt, 6 where c can be real or complex numbers. Then, formally we have a sum of general form ΦxEf0 c fx. 7 In certain cases, Φα + β or Φαβ can be decomposed into something having a power series in β as a part. Accordingly the operator ΦxE Φx + x Φxe D can be expressed as some power series involving operators or D s. Then it may be possible to compute the right-hand side of 7 by means of operator-series in or D s. This idea could be readily applied to various elementary functions Φt. Therefore, we can obtain various transformation formulas as well as series expansion formulas for the series of the form 7. It is well-nown that the Eulerian fraction is a powerful tool to study the Eulerian polynomial, Euler function and its generalization, Jordan function cf. [1]. The classical Eulerian fraction can be expressed in the form α m x where A m x is the mth degree Eulerian polynomial of the form A mx 1 x m+1 x 1, 8 A m x m!sm,x 1 x m, 9 4

5 Sm, being { Stirling } numbers of the second ind, i.e.,!sm, [ t m ] t0, which is also m denoted by. Evidently α m x can be written in the form cf. [4] α m x m!sm,x x +1 In order to express some new formulas for certain general types of power series, we need to introduce the following extension of Euler fraction, denoted by α n x,ax,bx,gx, using the generalized Sheffer-type polynomials associated with analytic functions or power series At, Bt, and gt, which satisfy conditions in Definition 1. α n x,ax,bx,gx : l! p l xsn,, 11 where p l x is the generalized Sheffer-type polynomial of degree l defined in Definition 1. In particular, if Ax 1 and gx x, then p l x B l 0x l /l!, and the generalized Euler fraction is hence because l α n x, 1,Bx,x l! p l x l Sn,B xx 12 l! l Obviously, α m x defined by 8 can be presented as B l 0 xl l! B xx. α m x A m x, 1, 1 x 1,x. From 11, two inds of generalized Eulerian fractions in terms of Ax 1, gx x, Bx 1 + x a and Bx 1 x a 1, respectively, with real number a as a parameter, can be introduced respectively, namely for x 1, and A n x, 1, 1 + x a,x a!sn,x x a A n x, 1, 1 x a 1,x a +!Sn,x 14 1 x a++1 for x 1. These two generalized Eulerian fractions were given in [5]. Another extension of the classical Eulerian fraction is presented in [10]. Two maor transformation and expansion formulas and their applications will be displayed in next section, and the convergence of the series in the formulas is presented in 3. 5

6 2 Series transformation-expansion formulas Theorem 2. Let {f} be a sequence of numbers in R or C, and let ht be infinitely differentiable. Assume At, Bt, and gt are analytic functions in a dis centered at the origin or power series defined as in Definition 1 and {p n x} is the generalized Sheffer-type polynomial sequence associated with Ax, Bx, and gx, we have formally fnp n x hnp n x l n f0 p l x n ln 1 1 hn 0α n x,ax,bx,gx, 2 where Sn, is the Stirling numbers of the second ind, and α n x,ax, Bx,gx n l! l pl xsn, is the generalized Eulerian fraction defined as in 11. In particular, if gt t, then the transformation and expansion formulas 1 becomes to fn n l l0 n f0 l0 A n l 0B l 0x l n A n l 1B l xx l. 3 l Proof. Applying the operator AEBxgE to ft at t 0, where E is the shift operator, we obtain the left-hand side of 1. On the other hand, we have AEBxgEft t0 A1 + Bxg1 + ft t0 p l x 1 + l ft l l t0 p l x n f0, n l0 l0 which implies the double sum on the right-hand side of 1. Similarly, for the infinitely differentiable function ht, we can present AEBxgEht t0 Ae D Bxge D ht t0 p l xe ld l n ht t0 p l x hn 0 l0 l0 p l xl n h n 0 l0 6

7 By applying 1 to the internal sum of the rightmost side of the above equation for ft t and noting S, t t0 /!, we obtain AEBxgEht t0 l p l x t n t0 l l p l x!sn, l h n 0 α n x,ax,bx,gx. h n 0 h n 0 This completes the proof of the theorem. If gt t, then we have formally p n x n l0 An l 0B l 0x l /n l!l! and AEBxEft t0 A1 + Bx + x ft t0 A n l 1 B l x x l ft n l! l! t0, l0 which can be written as the double sum on the right-hand side of 3. Remar 3. When ft and ht are polynomials, the right-hand sides of 1 and 2 are finite sums, which can be considered as the closed forms of the corresponding left-hand side series. For this reason, we call formulas 1 and 2 the series transformation and expansion or transformation-expansion formulas. Thus, for the rth degree polynomial φt, from 1 and 2 we have two expansion formulas, φnp n x φnp n x r l n φ0 p l x n r ln 4 1 φn 0α n x,ax,bx,gx, 5 where the right-hand sides can be considered as the GF s of {φnp n x}. Corollary 4. Let {α n x,ax,bx,gx} be the generalized Eulerian fraction sequence defined by 11. Then the exponential generating function of the sequence is Ae t Bxge t. In particular, the exponential generating function of sequence {α n x, 1,Bx,x} is Bxe t. Proof. The exponential GF of {α n x,ax,bx,gx} can be written as 7

8 α n x,ax,bx,gx tn l Sn, p l x tn l 1! u n l u0 p l x l t n. Applying formula 1 for f n into the double sum in the above parentheses yields α n x,ax,bx,gx tn n t n p x p xe t Ae t Bxge t. Here, the last step is due to Definition 1. We now give two special cases of Theorem 2. Corollary 5. Let {f} be a sequence of numbers in R or C, and let Bt and gt be infinitely differentiable on [0,. Then we have formally fnb n 0 xn gnb n 0 xn n f0b n x xn 6 g n 0 α n x, 1,Bx,x, 7 where α n x, 1,Bx,x is the generalized Eulerian fraction defined by 12. Proof. By setting At 1 and gt t into transformation-expansion formulas 1 and 2, we obtain formally p l x B l 0x l /l!. Thus, the modified formulas 1 and 2 are respectively 6 and 7. Example 6. Setting respectively Bt 1 t m 1 t 1 and Bt 1+t m t 1 into 6 and 7 yield the transformation-expansion formulas 8

9 m + fx m + hx m + x f0 1 x m++1 8 α x, 1, 1 x m 1,x D h0! 9 and m m fx hx m x f x m α x, 1, 1 + x m,x D h0, 11! respectively, where α x, 1, 1 x m 1,x and α x, 1, 1 + x m,x are defined in 14 and 13, respectively. By substituting m 0 in formulas 8 and 9 or applying transform x x and staing m 1 in formulas 10 and 11, we obtain fx hx x 1 x +1 f0 12 α x D h0, 13! where α x is defined by 8 and and 13 were shown as in [4]. And 12 is an extension of the following well-nown Euler series transform that can be found by setting x 1 into 12: 1 n fn 1 n 2 n+1 n f0. By applying operator E m and multiplying x m on the both sides of formula 8, we obtain its alternative form as follows: m fx m m + x +m fm 14 m 1 x m++1 Example 7. Let λ and θ be any real numbers. The generalized falling factorial t+λ θ p is usually defined by t + λ θ p Π p 1 t + λ θ, p 1, t + λ θ

10 It is nown that Howard s degenerate weighted Stirling numbers cf. [8] may be defined by the finite differences of t + λ θ p at t 0: Sp,,λ θ : 1 [ ] t + λ θ p! Then, using 14, 8, and 10 with ft t + λ θ p, we get t0. p m +!Sp,,λ θx + λ θ p x +m, 15 m 1 x m++1 m m + p m +!Sp,,λ θx + λ θ p x, 16 1 x m++1 p!sp,,λ θx m + λ θ p x The particular case of 17 with x 1, namely, m m + λ θ p p m x m m 2 m!sp,,λ θ, was given in formula 35 of [10], and the particular case of 16 with m 0 was considered in [9]. It is also obvious that the classical Euler s summation formula for the arithmeticgeometric series cf. for example, [3, Lemma 2.7] is implied by 16 with λ θ 0 and m 0, or by 17 with λ θ 0, m 1, x x. Some other series transformation-expansion formulas can be constructed formally from the above formulas by using integration or differentiation. For instance, taing the integral on the bother sides of 12 we obtain fx f0 ln1 x x f0, 18 1 x which can also be considered as a special case of 6 for Bt ln1 t. Using the substituting rule t D into 6 and applying the resulting operator to an infinitely differentiable function f with the similar argument shown in Theorem 2, we have ADBxgDft t0 p n xf n We now specify A, B and g in 19 to establish the following corollary. Corollary 8. If At,Bt,gt t/e t 1,e t,t, 2/e t +1,e t,t, t/lnt+1,e t, lnt+ 1, then from 19 we have 10

11 Dfx + y φ n xd n fy n n fx fx + y E n xd n f0 21 ψ n x n Dfy, 22 where φx and ψx are Bernoulli polynomials of the first and second ind, respectively, and E n x are Euler polynomials. Proof. If At,Bt,gt t/e t 1,e t,t, 2/e t + 1,e t,t, t/lnt + 1,e t, lnt + 1, then the corresponding Sheffer-type polynomials are p n x φ n x, E n x, and ψ n x, respectively cf. [11, pp. 250, 309, 279], and the corresponding operators on the left-hand side of 19 for the different At,Bt,gt become respectively De xd /e D 1 DE x /, 2e xd e D + 1 2Ex + 2 n 1 n, 2 and + 1 x /ln + 1 E x /D. Hence, the proof of the theorem is complete. The results in Corollary 8 were given in [7] by using different treatment for each individual formula while our method described here can be considered as a uniform approach, which can be used to find more transformation and expansion formulas. It is obvious that for x 0, formulas are specified as Dfy B 1 n 1 2 n n f0 fy Dn fy 23 e n D n f0 24 b n n Dfy, 25 where B n 1 φ n 0 is the first order generalized Bernoulli number, and e n E n 0, and b n ψ n 0. 3 Convergence of the series transformation-expansions As may be conceived, various formulas displayed in the list in Section 2 may be employed to construct some summation formulas with estimable remainders cf. the proof of Theorem 11

12 11 below. In what follows convergence problems related to the series expansions in Section 2 will be investigated. We now establish convergence conditions for the series expansions in 6 and 7. Suppose that {f} and {h} are bounded sequences say f < M and h < M for all, and that gz is analytic for z < ρ. Then it is follows that the left-hand sides of 6 and 7 as well as the right-hand sides of 6 and 7 are absolutely convergent series for x < ρ. Hence, we have the following convergence theorem. Theorem 9. If {f} and {g} are bounded sequences, and that Bz is analytic for z < ρ for some positive real number ρ, then the series expansions in 6 and 7 converge absolutely for all x < ρ. The convergence on the general case where {f} is not bounded presents some complicated situation. The next theorem gives a discussion for the series transformation-expansion formulas shown in 12, 13, and 18, and general way may be developed through it, which is left for the interested reader to consider. Theorem 10. Let {f} be a sequence of numbers in R or C, and denote θ : lim f 1/. Then the series expansions in 12, 13, and 18 are convergent for all nonzero x satisfying x θ < 1. Proof. Substituting the expression of α x defined by 10 into 12 and noting! S,D! e D 1 cf. [4] yields 13. More precisely, gx!! S, x x 1 x +1 α x D g0! g0 1 x +1D! S, D g0! x 1 x +1 g0. Hence, we only need to show the convergence of expansions in 12 and 18. In accordance with Cauchy s root test, the convergence of the series on the left-hand side of 12 and 18 is obvious because of the condition x θ < 1. To prove the convergence of the series expansion on the right-hand side of 12, we choose ρ > θ such that θ x < ρ x < 1. Thus for large we have f 1 < ρ. Consequently, f0 1 1 f 12 < 2 1 ρ ρ

13 as. Therefore, for every x 1/θ, x lim f0 1 x lim x 1 x f0 1 ρ x 1 x < ρ x < 1. Hence, from the root test, the series expansion on the right-hand side of 18 is convergent. Similarly, the expansion on the right-hand side of 12 converges as well. This completes the proof of the theorem. To extend the convergence intervals of the series expansions in 12 and 13, we need more precise estimation as follows. Theorem 11. Let {f} be a sequence of numbers in R or C, and let θ lim f 1/. Then for any given x with x 0 we have the convergent expressions 12 and 13 provided that x θ < 1. Proof. As we mentioned in the proof of Theorem 10, it is sufficient to show the convergence of 12. For this purpose, we now find a remainder of the expansion of 12 as follows. Formally, we have 1 xe 1 1 x x 1 1 x 1 1 x 1 1 x { n 1 l x x 1 x 1 l + n } 1 x 1 x 1 x l0 1 x n 1 x l n x n + 1 x l+1 l 1 x 1 xe l0 n 1 x l n x + x l E l n. 1 x l+1 l 1 x l0 Since E l n f0 n E l f0 n fl, applying operator 1 xe 1 and the rightmost operator shown above to ft t0, respectively, yields l0 1 xe 1 ft t0 n 1 fx x 1 x +1 f0 + x n 1 x n x l n fl. 1 l0 13

14 Since x θ < 1 x 0, the convergence of the series expansion on the left-hand side of 12 or 1 is obtain. To prove the convergence of the right-hand side of 1, i.e., the remainder form of 12, it is sufficient to show that l0 xl n fl is absolutely convergent. Choose ρ > θ such that θ x < ρ x < 1. Thus, for large we have f 1/ < ρ, i.e., f < ρ. Consequently we have, for large l as l. Thus n fl 1/l 1/l n fl + ρ1 + ρ n/l ρ lim x l n fl 1/l ρ x < 1, l 1/l n ρ l+ so that the series on the right-hand side of 1 or 12 is also convergent absolutely under the given conditions. A sequence {a n } is called a null sequence if for any given positive number ǫ, there exists and integer N such that every n > N implies a n < ǫ. [12] cf. Theorem 4 in Section 43 pointed out that a linear combination of {a n }, denoted by {a n n c n,a }, is also a null sequence if the coefficient set {c n, } 0 n n 0, 1, 2,... satisfies the following two conditions: i Every column contains a null sequence, i.e., for fixed 0, c n, 0 when n. ii There exists a constant K such that the sum a n,0 + a n,1 + + a n,n < K for every n. By using this claim of the null sequence, we can have the following convergence result of the series expansions in 12 and 13. Theorem 12. Suppose that {fn} is a given sequence of numbers real or complex such that fnxn is convergent for every x Ω with Ω, 0 φ. Then the series expressions on the right-hand sides of 12 and 13 converge for every x Ω, 0. Proof. We write the remainder of expression 1 as follows. 14

15 R n : x n 1 x n xn 1 x n xn 1 x n x n 1 x n l0 x l n fl l0 n 1 n x l n l0 1 x l n n x l f + l f + l x l fl xn 1 x n n x x, where x l xl fl 0 n. Since l0 xl fl converges, x is the term of a null sequence, applying the result on the linear combination of a null sequence shown above, we find the coefficients of x in the linear combination of the rightmost sum, c n, : xn n 1 x n x satisfy the following two conditions: 1 If is fixed, we have c n, 0 as n because c n, x n n n < 1 x n 1 x n and 1/1 x < 1 for every x Ω [ 1, 0; and n a n, x n n x < n 1 x n 1 x and x/1 x < 1 for every x Ω, 1. 2 For every n and for every x Ω, 0 we have a n, 1 1 x n n x n 1. Therefore, Theorem 4 in Section 43 of [12] shows that R n is also the term of a null sequence, so the series on the right-hand side of 12 converges for every x Ω, 0. In addition, the convergence of the right-hand series expansion of 13 is followed. We now discuss the convergence of the series expansions in Actually, we may sort the series transformation-expansion formulas associated with At, Bt, t into two classes. The first class includes only either the sum β D f or the sum γ E f in the formulas such as 20 and 22. The second class includes the sums β D f and/or γ E f on both sides of the transformation-expansion formulas lie 21. We may establish the following convergence theorem. 15

16 Theorem 13. For the first class series expansions associated with β D f or γ E f defined above, their absolute convergence are ensured if lim D f 1/ < 1 or lim E f 1/ < 1 and β 1 or γ 1. The second class series expansions defined above absolutely converge if lim D f 1/ < 1, β 1, and γ 1/e 1. Proof. The first half of the theorem is easy to be verified by using the root test. To prove the second half, we need the following statement: If f C, then lim D fy 1/ < a, a positive real number, implies lim fy 1/ < e a 1. 2 In fact, if we denote lim n D n fy 1/n θ, then there exists a number γ such that θ < γ < a. Thus for large enough n we have D n fy 1/n < γ or D n fy < γ n. Noting Sn,m 0 and D n fy < γ n, we obtain fy! Sn,Dn fy! Sn, Dn fy n n! Sn,γn e γ 1 < e a 1. n Here the rightmost equality is from Jordan [11, p. 176]. Therefore, lim D f 1/ < 1 implies that lim fy 1/ < e 1. Those two inequalities and the conditions for the coefficients {β } and {γ } confirm the absolute convergence of the second class series expansions with the root test. As a corollary of Theorem 13, we now establish the convergence results of the series expansions in Corollary 14. For given f C and x,y R, the absolute convergence of the series expansion 20 is ensured by the condition lim D fy 1/ < 1. 3 Similarly, the absolute convergence of the series expansion 21 and 22 are ensured by the conditions and respectively. lim lim Dfy 1/ < 1 4 D fy 1/ < 1, 5 16

17 Proof. From Theorem 13, it is sufficient to show that lim φ x 1/ 1, 6 and lim E x 1/ 1, 7 lim ψ x 1/ 1, 8 which will be proved below from the basic properties of φ x, E x, and ψ x shown as in Jordan [11]. Write the Bernoulli polynomials of the first ind, φ x, as cf. Jordan [11, p. 321] φ x x! α, where α B 1 /!, and B 1 are ordinary Bernoulli numbers. Note that α 0 1, α 1 1/2, α 2m+1 0 m N and cf. [11, p. 245] Thus for 2 α 2m 1 122π 2m 2, m 0, 1, 2,... φ x x! < + x 1 2 1! + x! e x. 2 1 x 122π 2! It follows that φ x 1/ < exp x / 1 as, which implies 6. Secondly, note that Euler polynomial E x can be written in the form E x e x!, e 0 1, 9 where e E 0, e 2m 0 m 1, 2,..., and e 2m 1 satisfies the inequality cf. [11, p. 302]. Thus we have the estimation e 2m 1 < 2 < 1 m 1, 2, π2m 2 E x x! + 1 e x! x! + 1 x! < e x. 17

18 Consequently we get lim E x 1/ lim e x 1/ 1. Hence 7 is verified. Finally, from [11, p. 268], we have an integral representation of ψ x, the Bernoulli polynomials of the second ind, namely 1 x + t ψ x dt. 11 For t [0, 1] and for large we have the estimation x + t x + t x t 1 + [ x ] o o [ x ].!!! This means that there is a constant M > 0 such that max x + t 0 t 1 < M [ x ]. Thus it follows that 1 lim ψ x 1/ lim 0 0 x + t dt This is a verification of 8, and corollary is proved. 1/ lim M [ x ] 1/ 1. Remar 15. The convergence conditions given in Theorem 10 can be improved by restricting f and using similar techniques shown in [4]. 4 Acnowledgments The author would lie to than the editor and the referee for their valuable suggestions and help. References [1] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, [2] G. P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translation of Math. Monographs, Vol. 59, American Mathematical Society, [3] H. W. Gould and J. Wetweerapong, Evaluation of some classes of binomial identities and two new sets of polynomials, Indian J. Math , [4] T. X. He, L. C. Hsu, P. J.-S. Shiue, and D. C. Torney, A symbolic operator approach to several summation formulas for power series, J. Comp. Appl. Math ,

19 [5] T. X. He, L. C. Hsu, and P. J.-S. Shiue, A symbolic operator approach to several summation formulas for power series II, [6] T. X. He, L. C. Hsu, and P. J.-S. Shiue, The Sheffer group and the Riordan group, Discrete Appl. Math , [7] T. X. He, L. C. Hsu, and P. J.-S. Shiue, Symbolization of generating functions, an application of Mullin-Rota s theory of binomial enumeration, Comp. Math. Appl , [8] F. T. Howard, Degenerate weighted Stirling numbers, Discrete Math , [9] L. C. Hsu and P. J.-S. Shiue, Cycle indicators and special functions, Ann. Combinatorics , [10] L. C. Hsu and P. J.-S. Shiue, On certain summation problems and generalizations of Eulerian polynomials and numbers, Discrete Math , [11] Ch. Jordan, Calculus of Finite Differences, Chelsea, New Yor, [12] K. Knopp, Theory and Application of Infinite Series, Hafner Publishing Comp., New Yor, [13] M. Petovše, H. S. Wilf, and D. Zeilberger, AB, AK Peters, [14] S. Roman, The Umbral Calculus, Academic Press, [15] S. Roman and G.-C. Rota, The umbral calculus, Advances in Math , [16] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically at nas/sequences/, [17] A. Sofo, Computational Techniques for the Summation of Series, Kluwer, [18] H. S. Wilf, Generatingfunctionology, 2nd Edition, Academic Press, Mathematics Subect Classification: Primary 41A58; Secondary 41A80, 65B10, 05A15, 33C45, 39A70. Keywords: Sheffer-type polynomials, symbolic operator, power series, transformationexpansion, generalized Eulerian fractions, Stirling number of the second ind. Concerned with sequences A000262, A052844, A084358, A129652, A129653, and A Received November ; revised version received April Published in Journal of Integer Sequences, July Return to Journal of Integer Sequences home page. 19