A symbolic operator approach to several summation formulas for power series II

Transcription

1 A sybolic operator approach to several suation forulas for power series II T. X. He, L. C. Hsu 2, and P. J.-S. Shiue 3 Departent of Matheatics and Coputer Science Illinois Wesleyan University Blooington, IL , USA 2 Departent of Matheatics, Dalian University of Technology Dalian 6024, P. R. China 3 Departent of Matheatical Sciences, University of Nevada, Las Vegas Las Vegas, NV , USA June 22, 2007 Abstract Here expounded is a ind of sybolic operator ethod that can be used to construct any transforation forulas and suation forulas for various types of power series including soe old ones and ore new ones. AMS Subect Classification: 05A5, 65B0, 33C45, 39A70, 4A80. Key Words and Phrases: sybolic operator, power series, generalized Eulerian fractions, Stirling nuber of the second ind. The research of this author was partially supported by Artistic and Scholarly Developent (ASD Grant and sabbatical leave of the IWU. This author would lie to than UNLV for a sabbatical leave during which the research in this paper was carried out.

2 2 T. X. He, L. C. Hsu, P. J.-S. Shiue Introduction - setch of an operator ethod As is well-nown, the closed for representation of series has been studied extensively. See, for exaples, Cotet [], Jordan [2], Egorychev [2], Roan-Rota [4], Sofo [5], Wilf [7], Petovše-Wilf-Zeilberger s boo [3], and the authors recent wor [8]. This paper is a sequel to the authors with Torney paper [6]. The obect of this paper is to ae use of the classical operators (difference, E (shift, and D (derivative to construct a ethod for the suation of power series that appears to have a certain wide scope of applications. An iportant tool used in the Calculus of Finite Differences and in Cobinatorial Analysis are the operators E,, and D defined by the following relations. Ef(t f(t +, f(t f(t + f(t, Df(t d dt f(t. Powers of these operators are defined in the usual way. In particular for any real nubers x, one ay define E x f(t f(t + x. Also, the nuber is defined as an identity operator, viz. f(t f(t. It is easy to verify that these operators satisfy the foral relations (cf. [2] E + e D, E e D, D log( +. Note that E f(0 [ E f(t ] f(, so that t0 (xe f(0 f(x. This eans that (xe with x as a paraeter ay be used to generate a general ter of the series f(x. Now suppose that Φ(t is an analytic function of t or a foral power series in t, say Φ(t c t, c [t ]Φ(t, (. where c can be either real or coplex nubers. have a su of general for Φ(xEf(0 Then, forally we c f(x. (.2 The operator Φ(xE Φ(x + x Φ(xe D can be expressed as soe power series involving operators or D s. Then it ay be possible to

3 Sybolic operator ethod to transforation and suation forulas3 copute the right-hand side of (.2 by eans of operator-series in or D s. This idea could be readily applied to various eleentary functions Φ(t. Indeed, if we tae Φ(t to be any of the following functions (i ( + t α, (ii ( t α, (iii e t (iv log( t, (v sin t, (vi cosh t, (.3 etc., thus, using suitable expressions of Φ(xE Φ(x + x Φ(xe D in ters of or D, we can obtain various transforation forulas as well as suation forulas for the series of the for (.2. Our results that will be presented in this paper are a significant iproveent of our previous wor with Torney shown in [6], in which the ain result is a special case of Theore 3.. Rear. Obviously, Φ(t is not liited to the functions shown as in (.3. For instance, we ay choose (cf. [0] Φ(t ( zt + yt λ P (, z, y, λt, (.4 the generating functions (GFs of the so-called Gegenbauer-Huberttype polynoials. As specific cases of (.4, we consider P (, z, y, λ as follows P (2, z,, U (z, Chebyshev 2nd ind polynoial, P (2, z,, /2 ψ (z, Legendre polynoial, P (2, z,, P + (z, P ell polynoial, P (2, z/2,, F + (z, F ibonacci polynoial, P (2, z/2, 2, Φ + (z, F erat st ind polynoial, where F + F + ( is the Fibonacci nuber. The expansion (.4 is a special case of the generalized Hubert polynoials studied by Gould in [3], in which a generalized Hubert polynoial P n (, x, y, p, C is defined by eans of (C xt + yt p t n P n (, x, y, p, C, n0

4 4 T. X. He, L. C. Hsu, P. J.-S. Shiue where is an integer and the other paraeters are unrestricted. In that paper, Gould first obtained soe recurrences satisfied by the P n and then gives a forula for D x P n+ that generalizes a forula of Catalan for the th derivative of the Legendre polynoial. He also showed that if the function f(x, t satisfies (td t f(x, t (x yt D x f(x, t, then where (td t r f(x, t Q r (, x, y, td x f(x, t (r, p p!( t p Q r p (, x, y, t n r t n P n (, x, y, p, xt yt n0 ( p r. Soe notations and an extension of Eulerian fractions will be given in next section. Two lists of transforation and suation forulas will be displayed in the latter Sections 3, and any illustrative exaples will be given in Section 4. 2 An extension of Eulerian fractions It is well-nown that the Eulerian fraction is a powerful tool in the study of the Eulerian polynoial, Euler function and its generalization, Jordan function (cf. []. The classical Eulerian fraction, α (x, can be expressed in the for α (x A (x ( x + (x, (2. where A (x is the th degree Eulerian polynoial of the for A (x 0 {! x ( x, (2.2 { { being Stirling nubers of the second ind, i.e.,! [ t ] t0. Evidently α (x can be written in the for (cf. [6]

5 Sybolic operator ethod to transforation and suation forulas5 α (x 0 {! x ( x. + In order to express soe new forulas for certain general types of power series, we need to introduce an extension of Euler fraction associated with an infinitely differentiable function g(x defined as A (x, g(x : 0 { g ( (xx, (2.3 where g ( (x is the -th derivative of g(x. Obviously, α (x defined by (2. can be presented as α (x A (x, ( x. Fro (2.3, two inds of generalized Eulerian fractions in ters of g(x ( + x α and g(x ( x α, with real nuber α as a paraeter, can be introduced respectively, naely A (x, α A (x, ( + x α A (x α ( + x { α (! x α, (x, (2.4 ( + x α 0 ( Ã (x, α A x, ( x α Ã (x α ( x { α++ (! x α +, (x. (2.5 ( x α++ 0 These ay be called, respectively, the st ind and 2nd ind of generalized Eulerian fractions. Correspondingly A (x, α and Ã(x, α are called the th degree generalized Eulerian polynoials, having explicit expressions as follows:

6 6 T. X. He, L. C. Hsu, P. J.-S. Shiue A (x, α Ã (x, α ( { α! 0 ( α +! 0 x ( + x (2.6 { x ( x. (2.7 As easily seen, Ã (x, 0 α (x A ( x,. 3 Series-transforation forulas All forulas presented in this section are foral identities in which we always assue that x or x according as ( x or ( + x appears in the forulas. Theore 3. Let {f( be a given sequence of nubers (real or coplex, and let g(t and h(t be infinitely differentiable on [0,. Then we have forally f(g ( (0 x! h(g ( (0 x! f(0g ( (x x! (3.! h( (0A (x, g(x, (3.2 where A (x, g(x is an extension of Euler fraction in ters of g(x defined as in (2.3. Proof. To prove (3., we apply the operator g(xe to f(t at t 0, where E is the shift operator. g(xef(t t0! g( (0 (xe f(t t0 f(g ( (0 x!. On the other hand, we have

7 Sybolic operator ethod to transforation and suation forulas7 g(xef(t t0 g(x + x f(t t0! g( (x (x f(t t0 f(0g ( (x x!. Siilarly, for the infinitely differentiable function h(t, we can present g(xeh(t t0 g(xe D h(t t0 0 x! g( (0! h( (0 0! g( (0 ( xe D h(t t0 ( g ( (0 x! 0! h( (0. By applying (3. to the inner su{ of the rightost side of the above equation for f(t t and noting ( t /!, we obtain t0 g(xeh(t t0 ( { 0 ( 0 g ( (xx! h( (0A (x, g(x. This copletes the proof of the theore. ( t t0 g( (x x!! h( (0! h( (0 Rear 3. The series transforation forulas (3. and (3.2 could have nuerous applications by setting different infinitely differentiable

8 8 T. X. He, L. C. Hsu, P. J.-S. Shiue functions for g(x. For exaples, we have f(x h(x ( α ( α + f(x x ( x + f(0 (g(x ( x, (3.3 α (x D h(0 (g(x ( x, (3.4! ( α x ( + x α f(0 (g(x ( + x α, (3.5 ( α + x ( x α++ f(0 (g(x ( x α, f(x (3.6 f(x e x x!! f(0 (g(x e x, (3.7 f(x ( x f(0 ln( x + f(0 (g(x ln( x, x (3.8 ( α h(x A (x, α D h(0 (g(x ( + x α, (3.9! ( α + h(x à (x, α D h(0 (g(x ( x α, (3.0! ( ( + f(x x + ( x ++ f( (g(x ( x, (3. ( h(x à (x, x D h( (g(x ( x, (3.2! ( f(2 + x 2+ ( x 2 2 f(0 sin x (2 +! (2! ( x f(0 + cos x (g(x sin x, (2 +! (3.3

9 Sybolic operator ethod to transforation and suation forulas9 ( f(2x 2 cos x (2! + sin x ( x 2 2 f(0 (2! ( + x f(0 (2 +! (g(x cos x, (3.4 f(2x 2 (2! ex 2 x f(0! + e x 2 ( x f(0,! (3.5 f(2 + x 2+ (2 +! ex 2 x f(0! e x 2 ( x f(0,! (3.6 where (3.5 and (3.6 are obtained by replacing g(x by e x and e x and adding and subtracting the resulting forulas respectively. Note that (3.3-(3.4 are well-nown and have been utilized to construct suation forulas with estiable reainders. See, e.g., He- Hsu-Shiue-Torney [6]. The particular cases of (3.5 with α (positive integer and (3.7 with f(x denoting a rth degree polynoial of x have been expounded in Probles (09 and (0 of Jolley s boo []. The rest of the above list appears to be not easily found in literature, and the forulas (3.9-(3.2 are believed to be new. Apparently, (3.3 is iplied by (3.5 (with α, x x and (3.6 (with α 0. Also, (3.4 is a particular case of (3.9 (with α, x x and (3.0 (with α 0. Moreover, it is easily observed that (3. and (3.2 can be derived fro (3.6 and (3.0, respectively, by substituting α, applying operator E, and ultiplying x on the both sides of the forer forulas. The transforation forulas given in the list are useful for accelerating convergence of power series because f(0 and D f(0 decreases to zero rapidly as ; e.g., the Euler series transforation and its extensions shown as in Proposition 3.2 of [6]. Rear 3.2 Fro (.4 we can derive Gegenbauer type series transforation forulas. For exaples, we consider

10 0 T. X. He, L. C. Hsu, P. J.-S. Shiue Φ(t ( 2zt + t 2 λ C (λ (zt, the GF of C (λ (z P (2, z,, λ, where P (2, z,, λ was shown as in (.4, and C ( (z U (z and C (/2 (z ψ (z are respectively the 2nd ind Chebyshev and Legengre polynoials. Using the sae arguent to derive (3.6 we obtain the following Gegenbauer type series transforation forula Φ(xEf(0 C (λ (zx f( ( ( λ + i λ + (z + δ i (z δ x i+ f(i + i i0 0 ( ( λ + i λ + ( 2zx + x 2 λ i i0 0 (z + δ i (z δ x i+ ( z + δx i ( (z δx i+ f(0. (3.7 (3.7 can also be verified directly as follows. By denoting δ z2 we can expand Φ(xE foral power series in ters of operator as Φ(xE ( (z + δxe λ ( (z δxe λ ( (z + δx (z + δx λ ( (z δx (z δx λ [ ] λ [ ] λ [ (z + δx] λ (z + δx [ (z δx] λ (z δx (z + δx (z δx ( ( λ + i λ + ( 2zx + x 2 λ i i0 0 (z + δi (z δ x i+ i+ ( z + δx i ( (z δx. Thus, (3.7 is obtained. In series transforation forula (3.7, we assue f(t to be a rth degree polynoial, denoted by φ(t, and obtain the generating function

11 Sybolic operator ethod to transforation and suation forulas { GF C (λφ( (z ( 2zx + x 2 λ i0 ( C (λ (zφ( x ( ( λ + i λ + 0 (z + δi (z δ x i+ i+ φ(0 ( z + δx i ( (z δx. (3.8 In particular, for λ and /2 we have generating functions GF {φ(u (z ( 2zx + x 2 i0 (z + δi (z δ x i+ i+ φ(0 ( z + δx i ( (z δx, GF {φ(ψ (z ( 2zx + x 2 /2 i i0 (z + δi (z δ x i+ i+ φ(0 ( z + δx i ( (z δx. ( ( λ + i λ + 0 i ( ( λ + i λ + Rear 3.3 Evidently, when f(t is a polynoial, all the forulas in Section 3 becoe closed for of suation forulas with a finite nuber of ters. Moreover, the Right-hand side of each forula ay also be viewed as a GF for the sequence of coefficeients contained in the power series on the left-hand side. Thus, for the rth degree polynoial φ(t, fro (3. and (3.2 we obtain two type GF s of {φ(g ( (0: 0 i φ(g ( (0 x! φ(g ( (0 x! φ(0g ( (x x!, (3.9! φ( (0A (x, g(x. (3.20 Replacing f and h by polynoial φ in (3.3-(3.6, we obtain the special cases of (3.9 and (3.20. For instance,

12 2 T. X. He, L. C. Hsu, P. J.-S. Shiue φ(x φ(x ( α ( α + φ(x x ( x + φ(0 (g(x ( x, (3.2 α (x D φ(0 (g(x ( x, (3.22! ( α x ( + x α φ(0 (g(x ( + x α, (3.23 ( α + x ( x α++ φ(0 (g(x ( x α, φ(x (3.24 φ(x e x x!! φ(0 (g(x e x, (3.25 φ(x ( x f(0 ln( x + φ(0 (g(x ln( x, x ( α φ(x ( α + ( φ(x φ(x ( φ(x (3.26 A (x, α D φ(0! (g(x ( + x α, (3.27 Ã (x, α D φ(0 (g(x ( x α, (3.28! ( + x + ( x ++ φ( (g(x ( x, (3.29 Ã (x, x D φ(! (g(x ( x. ( Illustrative exaples Certainly a great variety of special exaples could be given via applications of the forulas displayed in Section 3. In what follows we erely present soe selective exaples for references. Exaple 4. Taing α and x x in (3.5, we get (3.3,

13 Sybolic operator ethod to transforation and suation forulas3 which is a well-nown forula utilized in the constrction of a suation forula with a reainder in the recent paper by the authors [7] and the paper by authors with Torney (cf. [6]. Putting x in (3.3 and (3.8 we get ( f( ( f( ( f(0 2 + f(0 log 2 + ( f(0 2. These are nown as Euler s series transfor and its analogue, which ay soeties be used to construct slowly convergent series into rapidly convergent ones. Exaple 4.2 Fro (3.5 and noting (2. and (2.2, we obtain the su of the Euler s arithetic-geoetric series p x p x [ t p] t0 ( x + p { p! x α ( x + p (x, where α p (x is nown as Eulerian fraction (cf. [6]. Exaple 4.3 In (3.5 taing α n, f(t ( t, a th degree polynoial, so that f( (, we get n ( ( n ( n ν ν0 ( n x ν0 x ν ( + x ν n x ( + x n, ( ( n x ν t ν ν ( + x ν n ( t ( n ν ν t0 ν0 t0 x ν ( + x ν n δ ν where we use ( t r t0 ( t r t0 ( 0 r δr, the Kronecer sybol. This is (3.8 of Gould s boo [4].

14 4 T. X. He, L. C. Hsu, P. J.-S. Shiue Exaple 4.4 The series transforation forulas can be applied to construct a set of identities by substituting certain functions. Siilar to Exaple 4.3, taing f(t ( ( t r so that f( r in (3.6 yields (for x < ( α + r ( r x ( α + ( α + r r x ( x α++ x r ( x α+r+. ( t r t0 Consequently, ( α + r ( r 2 ( α + r r 2 α+. Siilarly, for f(t ( t r, (r N0 N {0, fro (3.7-(3.8 and (3.5-(3.6 we obtain respectively ( x xr ex r! r!, ( x r log( x + ( r x (r, r x ( 2 x 2 r (2! ex x r 2 r! + e x ( x r, 2 r! ( 2 + x 2+ r (2 +! ex x r 2 r! e x ( x r. 2 r! Exaple 4.5 In (3.6 taing f(t t r so that f( r, we get ( α + r x ( α + Forula (.26 in [4] can be written as x! ( x α++ { r.

15 Sybolic operator ethod to transforation and suation forulas5 n ( n r x ( + x n 0 ( n 0 ( n x ( + x ( ( x ( + x n [ t r] t0. This is obviously a particular case of forula (3.5 with α n and f(t t r. Exaple 4.6 Series transforation (3. can be used to extend Gould and Wetweerapong s coparable finite su forula (cf. [5, (24] to the infinite su setting, naely, ( p x ( + x + ( x ++ i0 r ( ( i (i + p. i Exaple 4.7 It is nown that the GF of Bell nubers W ( is W ( x! eex. Note that W ( is the nuber of all possible partition of a set with distinct eleents. Also, for g(x e x and f( W ( +, forula (3.9 iplies! W (x e x ( e x d dx e ex W ( + x! W ( + x+ ( +!! W (x. e x d ( e e x dx Coparing the coefficients of x in the leftost and the rightost expressions, we get W ( W (, which is called the Aiten identity (cf. Theore B in 5.4 of [].

16 6 T. X. He, L. C. Hsu, P. J.-S. Shiue Exaple 4.8 Our series transforation can be used to reconstruct the Dobinsi s forula (cf. [4a] in 5.4 of []. If g(t e t and f(t t r (r N, then (3.3 or (3.9 iplies This leads to e r x! ex r { r! { r x. W (r. Exaple 4.9 In (3.7-(3.0 and (3.5-(3.6, we substitute f(t t r and h(t t r and obtain respectively r x { r e x x,! r x f(0 log( x + ( α r x A r (x, α(r N {0, ( α + (2 r x 2 (2! r x Ãr(x, α(r N 0, ex 2 (2 + r x 2+ (2 +! ex 2 { r (! { r { x + e x r 2 { r x e x 2 ( x, x ( x, { r ( x. Exaple 4.0 In (3.5 taing f(t r t, (r > 0, r, so that f( r and f(0 0 ( ( r (r, we get ( α (rx ( α ((r x ( + x. α Siilarly, fro (3.6-(3.0 and (3.3-(3.6 we have respectively

17 Sybolic operator ethod to transforation and suation forulas7 ( α + (rx (rx e x ((r x,!! (rx f(0 log( x + ( α (rx ( α + ((r x ( x α++, A (x, α (ln r.! ( (r x, x Exaple 4. Recall that Bernoulli polynoials B n (t s are generated by the expression and enoy the properties e tx x e x n0 B n (t x n n! d dt B n(t nb n (t, (n, 2, with B 0 (t and B n (0 B n being called Bernoulli nubers. Note that D B n (t (n B n (t so that D B n (0 (n B n (0 (n B n, where (n are th falling factorial of n with step length. Now let g(x B n (x and f( r (n and r are integers with 0 r n Then, g ( (0 B n ( (0 (n B n and f ( (x (n B n (x, so that Theore 3. iplies n ( n B n r x { { 0 0 Since and 0 gives the well-nown expression B n (x ( { n r B n (x! x. (4. 0 for all, the particular case r 0 n ( n B n x.

18 8 T. X. He, L. C. Hsu, P. J.-S. Shiue Of course the right-hand side of (4. can be regarded as a GF of { ( n Bn r n. In addition, taing f(t B n (t in (3.9 and (3.0 respectively, we easily obtain ( α ( α + B n (x n ( n n B n (x A (x, αb n (4.2 ( n à (x, αb n. (4.3 Recalling that Ã(x, 0 α (x (the ordinary Eulerian fraction, we can find the last identity iplies (with α 0 B n (x n ( n α (xb n. (4.4 Surely siilar identities of soe interest ay be found for other classical special polynoials. Exaple 4.2 Let λ and θ be any real nubers. The generalized falling factorial (t + λ θ p is usually defined by (t + λ θ p Π p 0 (t + λ θ, (p, (t + λ θ 0. It is nown that Howard s degenerate weighted Stirling nubers (cf. [9] ay be defined by the finite differences of (t + λ θ p at t 0: S(p,, λ θ : [ ] (t + λ θ p!. t0 Then, using (3.23 and (3.24 with φ(t (t + λ θ p, we get ( α ( α + ( + λ θ p x ( + λ θ p x p ( α!s(p,, λ θx, (4.5 ( + x α p ( α +!S(p,, λ θx. (4.6 ( x α++ The particular case of (4.6 with α 0 was considered in [0]. It is also obvious that the classical Euler s suation forula for the aritheticgeoetric series (cf. for exaple, Lea 2.7 in [5] is iplied by (4.5

19 Sybolic operator ethod to transforation and suation forulas9 with λ θ 0, α, x x, or by (4.6 with λ θ 0 and α 0. Exaple 4.3 For any given positive integer denote φ(t ( t. It is easy to find that φ( ( t t (. Thus an application of (3.29 to ( t gives ( 2 x ( ( + This shows that the GF of the nuber sequence GF { ( 2 ( ( + x +. (4.7 ( x ++ { ( 2 is given by x +. (4.8 ( x ++ { ( Naturally one ay as to find GF 3. Actually, this can be wored out as follows. Let the left-hand side of (4.7 be Φ(x. Then using (4.7 we find ( 2 Φ(xEφ(0 (xe φ(0 ( ( + (xe + φ(0 ( xe ++ ( ( + x + ( x ++ ( ( + x + ( x ++ ( ( ( ( x x ( + 0 ( + + ( 3 x E + φ(0 ( x x x ++ ( x +++. φ( + Thus we obtain

20 20 T. X. He, L. C. Hsu, P. J.-S. Shiue { ( 3 GF ( ( ( ( + + x ++.(4.9 ( x +++ { ( A siilar process can be applied to find GF n for n 4, 5,. However, we have not yet nown the closed for of ( lx for general l. Exaple 4.4 Suppose that φ(t is an integral polynoial, naely, all its coefficients (including the constant ter are integers. It is easily seen that φ(0/! ( 0,, 2, are integers as well. [ In fact, each ter a t ( 0 of φ(t has a difference at zero: a t { { { ] t0 0 a! with and 0 ( >. So 0 φ(0/! is a linear cobination of Stirling nubers of 2nd ind with integer coefficients. Thus forula (3.25 iplies that φ(x /! is equal to e x ultiplying by an integral polynoial. In particular, for x, this iplies that φ(0 0! + φ(! + φ(2 2! + + φ(! + is an integral ultiple of e. Exaple 4.5 Every forula in Section 3 ay be used to yield a pair of related forulas involving the trigonoetric functions cos θ and sin θ. For instance, setting x ρe iθ ρ(cos θ + sin θ with ρ x > 0 and i 2, we can obtain a pair of forulas fro (3.24 as follows

21 Sybolic operator ethod to transforation and suation forulas2 ( α + φ(ρ cos θ ( α + ( α + ( α + φ(0re ( φ(ρ sin θ φ(0i ( (ρe iθ ( ρe iθ α++ (4.0 (ρe iθ, (4. ( ρe iθ α++ where Re(z and I(z denote respectively the real part and iaginary part of the coplex nuber z. Obviously (4.0 and (4. could be specialized in various ways. Rear 4. In this paper we have ostly considered the operator ethod for the cases when φ(t taes various eleentary functions. Fro Rears., 3.2, and 3.4, we can see that the ethod also apply to the cases where φ(t ay tae various suitable special functions. However, it still reains uch to be investigated. Acnowledgents. The authors would lie to than the referees and the editor for their valuable suggestions and help. References [] L. Cotet, Advanced Cobinatorics, Reidel, Dordrecht, 974. [2] G. P. Egorychev, Integral Representation and the Coputation of Cobinatorial Sus, Translation of Math. Monographs, Vol. 59, AMS, 984. [3] H. W. Gould, Inverse series relations and other expansions involving Hubert polynoials. Due Math. J , [4] H. W. Gould, Cobinatorial Identities, Revised Edition, Morgantown, W. Va., 972. [5] H. W. Gould and J. Wetweerapong, Evaluation of soe classes of binoial identities and two new sets of polynoials, Indian J. Math. 4(999, No. 2,

22 22 T. X. He, L. C. Hsu, P. J.-S. Shiue [6] T. X. He, L. C. Hsu, P. J.-S. Shiue, and D. C. Torney, A sybolic operator approach to several suation forulas for power series, J. Cop. Appl. Math., 77(2005, [7] T. X. He, L. C. Hsu, and P. J.-S. Shiue, On the convergence of the suation forulas constructed by using a sybolic operator approach, Coput. Math. Appl. 5 (2006, No. 3-4, [8] T. X. He, L. C. Hsu, and P. J.-S. Shiue, Sybolization of generating functions, an application of Mullin-Rota s theory of binoial enueration, Coput. Math. Appl., 2007, to appear. [9] F. T. Howard, Degenerate weighted Stirling nubers, Discrete Math. 57(985, No., [0] L. C. Hsu and P. J.-S. Shiue, Cycle indicators and special functions, Annals of Cobinatorics, 5(200, [] L. B. W. Jolley, Suation of series, 2d Revised Edition, Dover Publications, New Yor, 96. [2] Ch. Jordan, Calculus of Finite Differences, Chelsea, New Yor, 965. [3] M. Petovše, H. S. Wilf, and D. Zeilberger, AB, AK Peters, Ltd. Wellesley, Massachusetts, 996. [4] S. Roan and G.-C. Rota, The Ubral Calculus, Adv. in Math., 978, [5] A. Sofo, Coputational Techniques for the Suation of Series, Kluwer Acad., New Yor, [6] X.-H. Wang and L. C. Hsu, A suation forula for power series using Eulerian fractions, Fibonacci Quarterly 4 (2003, No., [7] H. S. Wilf, Generatingfunctionology, 2nd Edition, Acad. Press, New Yor, 994.