AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 871512 10.1155/2013/871512 871512 Research Article Umbral Calculus and the Frobenius-Euler Polynomials Kim Dae San 1 Kim Taekyun 2 Lee Sang-Hun 3 Trujillo Juan J. 1 Department of Mathematics Sogang University, Seoul 121-742 Republic of Korea sogang.ac.kr 2 Department of Mathematics Kwangwoon University, Seoul 139-701 Republic of Korea kw.ac.kr 3 Division of General Education Kwangwoon University, Seoul 139-701 Republic of Korea kw.ac.kr 2013 7 2 2013 2013 27 11 2012 19 12 2012 2013 Copyright © 2013 Dae San Kim et al. 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 study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

1. Introduction

Let C be the complex number field. For λC with λ1, the Frobenius-Euler polynomials are defined by the generating function to be (1)1-λet-λext=eH(xλ)t=n=0Hn(xλ)tnn!, (see ) with the usual convention about replacing Hn(xλ) by Hn(xλ).

In the special case, x=0,Hn(0λ)=Hn(λ) are called the nth Frobenius-Euler numbers. By (1), we get (2)Hn(xλ)=l=0n(nl)Hn-l(λ)xl=(H(λ)+x)n, (see ) with the usual convention about replacing Hn(λ) by Hn(λ).

Thus, from (1) and (2), we note that (3)(H(λ)+1)n-λHn(λ)=(1-λ)δ0,n, where δn,k is the kronecker symbol (see [1, 10, 11]).

For rZ+, the Frobenius-Euler polynomials of order r are defined by the generating function to be (4)(1-λet-λ)rext=(1-λet-λ)××(1-λet-λ)extr-times=n=0Hn(r)(xλ)tnn!. In the special case, x=0,Hn(r)(0λ)=Hn(r)(λ) are called the nth Frobenius-Euler numbers of order r (see [1, 10]).

From (4), we can derive the following equation: (5)Hn(r)(xλ)=l=0n(nl)Hn-l(r)(λ)xl,Hn(r)(λ)=l1++lr=n(nl1,,lr)Hl1(λ)Hlr(λ). By (5), we see that Hn(r)(xλ) is a monic polynomial of degree n with coefficients in Q(λ).

Let be the algebra of polynomials in the single variable x over C and let * be the vector space of all linear functionals on . As is known, Lp(x) denotes the action of the linear functional L on a polynomial p(x) and we remind that the addition and scalar multiplication on * are, respectively, defined by (6)L+Mp(x)=Lp(x)+Mp(x),cLp(x)=cLp(x), where c is a complex constant (see [3, 12]).

Let F denote the algebra of formal power series: (7)F={f(t)=k=0akk!tkakC} (see [3, 12]). The formal power series define a linear functional on by setting (8)f(t)xn=an,n0. Indeed, by (7) and (8), we get (9)tkxn=n!δn,k(n,k0) (see [3, 12]). This kind of algebra is called an umbral algebra.

The order O(f(t)) of a nonzero power series f(t) is the smallest integer k for which the coefficient of tk does not vanish. A series f(t) for which O(f(t))=1 is said to be an invertible series (see [2, 12]). For f(t),g(t)F, and p(x), we have (10)f(t)g(t)p(x)=f(t)g(t)p(x)=g(t)f(t)p(x) (see ). One should keep in mind that each f(t)F plays three roles in the umbral calculus: a formal power series, a linear functional, and a linear operator. To illustrate this, let p(x) and f(t)=eytF. As a linear functional, eyt satisfies eytp(x)=p(y). As a linear operator, eyt satisfies eytp(x)=p(x+y) (see ). Let sn(x) denote a polynomial in x with degree n. Let us assume that f(t) is a delta series and g(t) is an invertible series. Then there exists a unique sequence sn(x) of polynomials such that g(t)f(t)ksn(x)=n!δn,k for all n,k0 (see [3, 12]). This sequence sn(x) is called the Sheffer sequence for (g(t),f(t)) which is denoted by sn(x)~(g(t),f(t)). If sn(x)~(1,f(t)), then sn(x) is called the associated sequence for f(t). If sn(x)~(g(t),t), then sn(x) is called the Appell sequence.

Let sn(x)~(g(t),f(t)). Then we see that (11)h(t)=k=0h(t)sk(x)k!g(t)f(t)k,      h(t)F,p(x)=k=0g(t)f(t)kp(x)k!sk(x),      p(x),f(t)sn(x)=nsn-1(x),f(t)p(αx)=f(αtp(x),(12)1g(f-(t))eyf-(t)=k=0sk(y)k!tk,yC, where f-(t) is the compositional inverse of f(t) (see ). In this paper, we study some properties of umbral calculus related to the Appell sequence. For those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

2. The Frobenius-Euler Polynomials and Umbral Calculus

By (4) and (12), we see that (13)Hn(r)(xλ)~((et-λ1-λ)r,t). Thus, by (13), we get (14)(et-λ1-λ)rtkHn(r)(xλ)=n!δn,k. Let (15)n(λ)={p(x)Q(λ)[x]degp(x)n}. Then it is an (n+1)-dimensional vector space over Q(λ).

So we see that {H0(r)(xλ),H1(r)(xλ),,Hn(r)(xλ)} is a basis for n(λ). For p(x)n(λ), let (16)p(x)=k=0nCkHk(r)(xλ),      (n0). Then, by (13), (14), and (16), we get (17)(et-λ1-λ)rtkp(x)=l=0nCl(et-λ1-λ)rtkHl(r)(xλ)=l=0nCll!δl,k=k!Ck. From (17), we have (18)Ck=1k!(et-λ1-λ)rtkp(x)=1k!(et-λ1-λ)rDkp(x)=1k!(1-λ)rj=0r(rj)(-λ)r-jejtDkp(x)=1k!(1-λ)rj=0r(rj)(-λ)r-jt0ejtDkp(x)=1k!(1-λ)rj=0r(rj)(-λ)r-jt0Dkp(x+j). Therefore, by (16) and (18), we obtain the following theorem.

Theorem 1.

For p(x)n(λ), let (19)p(x)=k=0nCkHk(r)(x). Then one has (20)Ck=1k!(1-λ)rj=0r(rj)(-λ)r-jDkp(j), where Dp(x)=dp(x)/dx.

From Theorem 1, we note that (21)p(x)=1(1-λ)r·k=0n{j=0r1k!(rj)(-λ)r-jDkp(j)}Hk(r)(xλ). Let us consider the operator Δ~λ with Δ~λf(x)=f(x+1)-λf(x) and let Jλ=(1/(1-λ))Δ~λ. Then we have (22)Jλ(f)(x)=11-λ{f(x+1)-λf(x)}. Thus, by (22), we get (23)Jλ(Hn(r)(xλ))=11-λ{Hn(r)(x+1λ)-λHn(r)(xλ)}. From (4), we can derive (24)n=0{Hn(r)(x+1λ)-λHn(r)(xλ)}tnn!=(1-λet-λ)re(x+1)t-λ(1-λet-λ)rext=(1-λet-λ)rext(et-λ)=(1-λ)(1-λet-λ)r-1ext=(1-λ)n=0Hn(r-1)(xλ)tnn!. By (23) and (24), we get (25)Jλ(Hn(r)(xλ))=Hn(r-1)(xλ). From (25), we have (26)Jλr(Hn(r)(xλ))=Jλr-1(Hn(r-1)(xλ))==Hn(0)(xλ)=xn,Jλr(xn)=JλrHn(0)(xλ)=Hn(-r)(xλ)=Jλ2rHn(r)(xλ). For sZ+, from (25), we have (27)Jλs(Hn(r)(xλ))=Hn(r-s)(xλ). On the other hand, by (12), (13), and (25), (28)Jλs(Hn(r)(xλ))=(et-λ1-λ)s(Hn(r)(xλ))=1(1-λ)s((1-λ)+k=1tkk!)s·(Hn(r)(xλ)). Thus, by (28), we get (29)Jλs(Hn(r)(xλ))=m=0s(sm)(1-λ)ml=m(k1++km=lkj11k1!km!)tl(Hn(r)(xλ))=m=0s(sm)(1-λ)ml=m1l!(k1++km=lkj1(lk1,,km)Dl)·Hn(r)(xλ)=m=0min{s,n}(sm)(1-λ)ml=mn(nl)k1++km=lkj1(lk1,,km)Hn-l(r)(xλ)=l=0min{s,n}{(nl)m=0l(sm)(1-λ)ml=0min{s,n}·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ)+l=min{s,n}+1n{(nl)m=0min{s,n}(sm)(1-λ)mm=0l=min{s,n}+1·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ). Therefore, by (27) and (29), we obtain the following theorem.

Theorem 2.

For any r,s0, one has (30)Hn(r-s)(xλ)=l=0min{s,n}{(nl)m=0l(sm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{s,n}+1n{(nl)m=0min{s,n}(sm)(1-λ)mbbbbbbbbb·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ).

Let us take s=r-1(r1) in Theorem 2. Then we obtain the following corollary.

Corollary 3.

For n0,r1, one has (31)Hn(xλ)=l=0min{r-1,n}{(nl)m=0l(r-1m)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{r-1,n}+1n{(nl)m=0min{r-1,n}(r-1m)(1-λ)m·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ).

Let us take s=r(r1) in Theorem 2. Then we obtain the following corollary.

Corollary 4.

For n0,r1, one has (32)xn=l=0min{r,n}{(nl)m=0l(rm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{r,n}+1n{(nl)m=0min{r,n}(rm)(1-λ)ml=min{r,n}+1n·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ).

Now, we define the analogue of Stirling numbers of the second kind as follows: (33)Sλ(n,k)=1k!j=0k(kj)(-λ)k-jjn,(n,k0). Note that S1(n,k)=S(n,k) is the Stirling number of the second kind.

From the definition of Δ~λ, we have (34)Δ~λnf(0)=k=0n(nk)(-λ)n-kf(k). By (33) and (34), we get (35)Sλ(n,k)=1k!Δ~λk0n,(n,k0). Let us take s=2r. Then we have (36)Jλrxn=Hn(-r)(xλ)=l=0min{2r,n}{(nl)m=0l(2rm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{2r,n}+1n{(nl)m=0min{2r,n}(2rm)(1-λ)ml=min{2r,n}+1n·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ),Jλrxn=(11-λΔ~λ)rxnJλrxn=1(1-λ)rj=0r(rj)(-λ)r-j(x+j)n. By (36), we get (37)1(1-λ)rj=0r(rj)(-λ)r-j(x+j)n=1(1-λ)rΔ~λrxn=l=0min{2r,n}{(nl)m=0l(2rm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{2r,n}+1n{(nl)m=0min{2r,n}(2rm)(1-λ)ml=min{2r,n}+1n·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ). Let us take x=0 in (37). Then we obtain the following theorem.

Theorem 5.

We have (38)r!(1-λ)rSλ(n,r)=r!(1-λ)rΔ~λr0nr!=l=0min{2r,n}{(nl)m=0l(2rm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(λ)+l=min{2r,n}+1n{(nl)m=0min{2r,n}(2rm)(1-λ)m·k1++km=lkj1(lk1,,km)}Hn-l(r)(λ)=m=0min{r,n}(rm)(1-λ)mk1++km=nkj1(nk1,,km).

Let us consider s=2r-1 in the identity of Theorem 2. Then we have (39)Jλr-1xn=Hn-(r-1)(xλ)=l=0min{2r-1,n}{(nl)m=0l(2r-1m)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(xλ)+l=min{2r-1,n}+1n{(nl)m=0min{2r-1,n}(2r-1m)(1-λ)m+l=min{2r-1,n}+1n·k1++km=lkj1(lk1,,km)}Hn-l(r)(xλ)=1(1-λ)r-1j=0r-1(r-1j)(-λ)r-1-j(x+j)n=1(1-λ)r-1Δ~λr-1xn. Let us take x=0 in (39). Then we obtain the following theorem.

Theorem 6.

For n0 and r1, one has (40)(r-1)!(1-λ)r-1Sλ(n,r-1)=(r-1)!(1-λ)r-1Δ~λr-10n(r-1)!=l=0min{2r-1,n}{(nl)m=0l(2r-1m)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(r)(λ)+l=min{2r-1,n}+1n{(nl)m=0min{2r-1,n}(2r-1m)(1-λ)ml=min{2r-1,n}+1n·k1++km=lkj1(lk1,,km)}Hn-l(r)(λ).

Remark 7.

Note that (41)(r-1)!(1-λ)r-1Sλ(n,r-1)=l=0min{r,n}{(nl)m=0l(rm)(1-λ)mk1++km=lkj1(lk1,,km)}·Hn-l(λ)+l=min{r,n}+1n{(nl)m=0min{r,n}(rm)(1-λ)m·k1++km=lkj1(lk1,,km)}Hn-l(λ).

Acknowledgment

The authors would like to express their gratitude to the referees for their valuable suggestions.

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