Apostol-Euler polynomials arising from umbral calculus

Abstract

In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-Euler polynomials.

MSC:05A40.

1 Introduction

Let ${\mathrm{\Pi }}_n$ be the set of all polynomials in a single variable x over the complex field ℂ of degree at most n. Clearly, ${\mathrm{\Pi }}_n$ is a $(n+1)$-dimensional vector space over ℂ. Define

$$\mathcal{H}=\{f(t)=\sum _{k\ge 0}{a}_k\frac{t^k}{k!}|a_k\in \mathbb{C}\}$$

(1.1)

to be the algebra of formal power series in a single variable t. As is known, $〈L|p(x)〉$ denotes the action of a linear functional $L\in \mathcal{H}$ on a polynomial $p(x)$, and we remind that the vector space on ${\mathrm{\Pi }}_n$ is defined by

$$〈cL+c^{\prime }{L}^{\prime }|p(x)〉=c〈L|p(x)〉+c^{\prime }〈L^{\prime }|p(x)〉$$

for any $c,c^{\prime }\in \mathbb{C}$ and $L,L^{\prime }\in \mathcal{H}$ (see [1–4]). The formal power series in variable t define a linear functional on ${\mathrm{\Pi }}_n$ by setting

$$〈f(t)|x^n〉=a_n\text{for all }n\ge 0\text{(see [1–4])}.$$

(1.2)

By (1.1) and (1.2), we have

$$〈t^k|x^n〉=n!{\delta }_{n,k}\text{for all }n,k\ge 0\text{(see [1–4])},$$

(1.3)

where ${\delta }_{n,k}$ is the Kronecker symbol. Let $f_L(t)={\sum }_{k\ge 0}〈L|x^k〉\frac{t^k}{k!}$ with $L\in \mathcal{H}$. From (1.3), we have $〈f_L(t)|x^n〉=〈L|x^n〉$. So, the map $L\mapsto f_L(t)$ is a vector space isomorphic from ${\mathrm{\Pi }}_n$ onto ℋ. Henceforth, ℋ is thought of as a set of both formal power series and linear functionals. We call ℋ umbral algebra. The umbral calculus is the study of umbral algebra.

Let $f(t)\in \mathcal{H}$. The smallest integer k for which the coefficient of $t^k$ does not vanish is called the order of $f(t)$ and is denoted by $O(f(t))$ (see [1–4]). If $O(f(t))=1$, $O(f(t))=0$, then $f(t)$ is called a delta, an invertible series, respectively. For given two power series $f(t),g(t)\in \mathcal{H}$ such that $O(f(t))=1$ and $O(g(t))=0$, there exists a unique sequence $S_n(x)$ of polynomials with $〈g(t){(f(t))}^k|S_n(x)〉=n!{\delta }_{n,k}$ (this condition sometimes is called orthogonality type) for all $n,k\ge 0$. The sequence $S_n(x)$ is called the Sheffer sequence for $(g(t),f(t))$ which is denoted by $S_n(x)\sim (g(t),f(t))$ (see [1–4]).

For $f(t)\in \mathcal{H}$ and $p(x)\in \mathrm{\Pi }$, we have

$$〈e^{yt}|p(x)〉=p(y),〈f(t)g(t)|p(x)〉=〈f(t)|g(t)p(x)〉,$$

(1.4)

and

$$f(t)=\sum _{k\ge 0}〈f(t)|x^k〉\frac{t^k}{k!},p(x)=\sum _{k\ge 0}〈t^k|p(x)〉\frac{x^k}{k!}$$

(1.5)

(see [1–4]). From (1.5), we derive

$$〈t^k|p(x)〉=p^{(k)}(0),〈1|p^{(k)}(x)〉=p^{(k)}(0),$$

(1.6)

where $p^{(k)}(0)$ denotes the k th derivative of $p(x)$ with respect to x at $x=0$. Let $S_n(x)\sim (g(t),f(t))$. Then we have

$$\frac{1}{g(\overline{f}(t))}{e}^{y\overline{f}(t)}=\sum _{k\ge 0}{S}_k(y)\frac{t^k}{k!},$$

(1.7)

for all $y\in \mathbb{C}$, where $\overline{f}(t)$ is the compositional inverse of $f(t)$ (see [1–6]).

For $\lambda \in \mathbb{C}$ with $\lambda \ne -1$, the Apostol-Euler polynomials (see [7–10]) are defined by the generating function to be

$$\frac{2}{\lambda e^t+1}{e}^{xt}=\sum _{k\ge 0}{E}_k(x|\lambda )\frac{t^k}{k!}.$$

(1.8)

In particular, $x=0$, $E_n(0|\lambda )=E_n(\lambda )$ is called the nth Apostol-Euler number. From (1.8), we can derive

$$E_n(x|\lambda )=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)E_{n-k}(\lambda )x^k.$$

(1.9)

By (1.9), we have $\frac{d}{dx}{E}_n(x|\lambda )=n{E}_{n-1}(x|\lambda )$. Also, from (1.8) we have

$$\frac{2}{\lambda e^t+1}=e^{E(\lambda )t}=\sum _{n\ge 0}{E}_n(\lambda )\frac{t^n}{n!}$$

(1.10)

with the usual convention about replacing $E^n(\lambda )$ by $E_n(\lambda )$. By (1.10), we get

$$2=e^{E(\lambda )t}(\lambda e^t+1)=\lambda e^{(E(\lambda )+1)t}+e^{E(\lambda )t}=\sum _{n\ge 0}(\lambda {(E(\lambda )+1)}^n+E_n(\lambda ))\frac{t^n}{n!}.$$

Thus, by comparing the coefficients of the both sides, we have

$$\lambda {(E(\lambda )+1)}^n+E_n(\lambda )=2{\delta }_{n,0}.$$

(1.11)

As is well known, the Bernoulli polynomial (see [11–14]) is also defined by the generating function to be

$$\frac{t}{e^t-1}{e}^{xt}=\sum _{k\ge 0}{B}_k(x)\frac{t^k}{k!}.$$

(1.12)

In the special case, $x=0$, $B_n(0)=B_n$ is called the nth Bernoulli number. By (1.12), we get

$$B_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_{n-k}{x}^k.$$

(1.13)

From (1.12), we note that

$$\frac{t}{e^t-1}=e^{Bt}=\sum _{n\ge 0}{B}_n\frac{t^n}{n!}$$

(1.14)

with the usual convention about replacing $B^n$ by $B_n$. By (1.13) and (1.14), we get

$$t=e^{Bt}(e^t-1)=e^{(B+1)t}-e^{Bt}=\sum _{n\ge 0}({(B+1)}^n-B_n)\frac{t^n}{n!},$$

which implies

$$B_n(1)-B_n={(B+1)}^n-B_n={\delta }_{n,1},B_0=1.$$

(1.15)

Euler polynomials (see [4, 11, 13, 15]) are defined by

$$\frac{2}{e^t+1}{e}^{xt}=\sum _{k\ge 0}{E}_k(x)\frac{t^k}{k!}.$$

(1.16)

In the special case, $x=0$, $E_n(0)=E_n$ is called the nth Euler number. By (1.16), we get

$$\frac{2}{e^t+1}=e^{Et}=\sum _{n\ge 0}{E}_n\frac{t^n}{n!}$$

(1.17)

with the usual convention about replacing $E^n$ by $E_n$. By (1.16) and (1.17), we get

$$2=e^{Et}(e^t+1)=e^{(E+1)t}+e^{Et}=\sum _{n\ge 0}({(E+1)}^n+E_n)\frac{t^n}{n!},$$

which implies

$$E_n(1)+E_n={(E+1)}^n+E_n=2{\delta }_{n,0}.$$

(1.18)

For $\lambda \in \mathbb{C}$ with $\lambda \ne -1$, the Frobenius-Euler (see [11, 16–19]) polynomials are defined by

$$\frac{1+\lambda }{e^t+\lambda }{e}^{xt}=\sum _{k\ge 0}{F}_k(x|-\lambda )\frac{t^k}{k!}.$$

(1.19)

In the special case, $x=0$, $F_n(0|-\lambda )=F_n(-\lambda )$ is called the nth Frobenius-Euler number (see [17]). By (1.19), we get

$$\frac{1+\lambda }{e^t+\lambda }=e^{Ft}=\sum _{n\ge 0}{F}_n(-\lambda )\frac{t^n}{n!}$$

(1.20)

with the usual convention about replacing $F^n(-\lambda )$ by $F_n(-\lambda )$ (see [17]). By (1.19) and (1.20), we get

$$1+\lambda =e^{F(-\lambda )t}(e^t+\lambda )=e^{(F(-\lambda )+1)t}+\lambda e^{F(-\lambda )t}=\sum _{n\ge 0}({(F(-\lambda )+1)}^n+\lambda F_n(-\lambda ))\frac{t^n}{n!},$$

which implies

$$\lambda F_n(-\lambda )+F_n(1|-\lambda )=\lambda F_n(-\lambda )+{(F(-\lambda )+1)}^n=(1+\lambda ){\delta }_{n,0}.$$

(1.21)

In the next section, we present our main theorem and its applications. More precisely, by using the orthogonality type, we write any polynomial in ${\mathrm{\Pi }}_n$ as a linear combination of the Apostol-Euler polynomials. Several applications related to Bernoulli, Euler and Frobenius-Euler polynomials are derived.

2 Main results and applications

Note that the set of the polynomials $E_0(x|\lambda ),E_1(x|\lambda ),\dots ,E_n(x|\lambda )$ is a good basis for ${\mathrm{\Pi }}_n$. Thus, for $p(x)\in {\mathrm{\Pi }}_n$, there exist constants $c_0,c_1,\dots ,c_n$ such that $p(x)={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$. Since $E_n(x|\lambda )\sim ((1+\lambda e^t)/2,t)$ (see (1.7) and (1.8)), we have

$$〈\frac{1+\lambda e^t}{2}{t}^k|E_n(x|\lambda )〉=n!{\delta }_{n,k},$$

which gives

$$〈\frac{1+\lambda e^t}{2}{t}^k|p(x)〉=\sum _{\ell =0}^n{c}_{\ell }〈\frac{1+\lambda e^t}{2}{t}^k|E_{\ell }(x|\lambda )〉=\sum _{\ell =0}^n{c}_{\ell }\ell !{\delta }_{\ell ,k}=k!c_k.$$

Hence, we can state the following result.

Theorem 2.1 For all $p(x)\in {\mathrm{\Pi }}_n$, there exist constants $c_0,c_1,\dots ,c_n$ such that $p(x)={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$c_k=\frac{1}{2k!}〈(1+\lambda e^t)t^k|p(x)〉.$$

Now, we present several applications for our theorem. As a first application, let us take $p(x)=x^n$ with $n\ge 0$. By Theorem 2.1, we have $x^n={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$c_k=\frac{1}{2k!}〈(1+\lambda e^t)t^k|x^n〉=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)〈1+\lambda e^t|x^{n-k}〉=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)({\delta }_{n-k,0}+\lambda ),$$

which implies the following identity.

Corollary 2.2 For all $n\ge 0$,

$$x^n=\frac{1}{2}{E}_n(x|\lambda )+\frac{\lambda }{2}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)E_k(x|\lambda ).$$

Let $p(x)=B_n(x)\in {\mathrm{\Pi }}_n$, then by Theorem 2.1 we have that $B_n(x)={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$\begin{array}{rl}{c}_k& =\frac{1}{2k!}〈(1+\lambda e^t)t^k|B_n(x)〉=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)〈1+\lambda e^t|B_{n-k}(x)〉\\ =\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(B_{n-k}+\lambda B_{n-k}(1)),\end{array}$$

which, by (1.15), implies the following identity.

Corollary 2.3 For all $n\ge 2$,

$$B_n(x)=\frac{(\lambda -1)n}{4}{E}_{n-1}(x|\lambda )+\frac{1+\lambda }{2}\sum _{k=0,k\ne n-1}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)B_{n-k}{E}_k(x|\lambda ).$$

Let $p(x)=E_n(x)$, then by Theorem 2.1 we have that $E_n(x)={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$\begin{array}{rl}{c}_k& =\frac{1}{2k!}〈(1+\lambda e^t)t^k|E_n(x)〉=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)〈1+\lambda e^t|E_{n-k}(x)〉\\ =\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(E_{n-k}+\lambda E_{n-k}(1)),\end{array}$$

which, by (1.18), implies the following identity.

Corollary 2.4 For all $n\ge 0$,

$$E_n(x)=\frac{1+\lambda }{2}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)E_{n-k}{E}_k(x|\lambda ).$$

For another application, let $p(x)=F_n(x|-\lambda )$, then by Theorem 2.1 we have that $F_n(x|-\lambda )={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$\begin{array}{rl}{c}_k& =\frac{1}{2k!}〈(1+\lambda e^t)t^k|F_n(x|-\lambda )〉=\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)〈1+\lambda e^t|F_{n-k}(x|-\lambda )〉\\ =\frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{k}\right)(F_{n-k}(-\lambda )+\lambda F_{n-k}(1|-\lambda )),\end{array}$$

which, by (1.21), implies the following identity.

Corollary 2.5 For all $n\ge 1$,

$$F_n(x|-\lambda )=\frac{1+\lambda }{2}{E}_n(x|\lambda )+\frac{1-{\lambda }^2}{2}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)F_{n-k}(-\lambda )E_k(x|\lambda ).$$

Again, let $p(x)=y_n(x)={\sum }_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\frac{x^k}{2^k}$ be the n th Bessel polynomial (which is the solution of the following differential equation $x^2{f}^{″}(x)+2(x+1)f^{\prime }+n(n+1)f=0$, where $f^{\prime }(x)$ denotes the derivative of $f(x)$, see [3, 4]). Then, by Theorem 2.1, we can write $y_n(x)={\sum }_{k=0}^n{c}_k{E}_k(x|\lambda )$, where

$$\begin{array}{rl}{c}_k& =\frac{1}{2k!}\sum _{\ell =0}^n\frac{(n+\ell )!}{(n-\ell )!\ell !2^{\ell }}〈1+\lambda e^t|t^k{x}^{\ell }〉\\ =\frac{1}{2}\sum _{\ell =k}^n\frac{(n+\ell )!}{(n-\ell )!\ell !2^{\ell }}\left(\genfrac{}{}{0ex}{}{\ell }{k}\right)〈1+\lambda e^t|x^{\ell -k}〉\\ =\frac{1}{2}\sum _{\ell =k}^n\frac{(n+\ell )!}{(n-\ell )!\ell !2^{\ell }}\left(\genfrac{}{}{0ex}{}{\ell }{k}\right)({\delta }_{n-k,0}+\lambda )\\ =\frac{k!}{2^{k+1}}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)+\lambda \sum _{\ell =k}^n\frac{k!}{2^{\ell +1}}\left(\genfrac{}{}{0ex}{}{\ell }{k}\right)\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\left(\genfrac{}{}{0ex}{}{n+\ell }{\ell }\right),\end{array}$$

which implies the following identity.

Corollary 2.6 For all $n\ge 1$,

$$y_n(x)=\sum _{k=0}^n\frac{k!}{2^{k+1}}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)E_k(x|\lambda )+\lambda \sum _{k=0}^n\sum _{\ell =k}^n\frac{k!}{2^{\ell +1}}\left(\genfrac{}{}{0ex}{}{\ell }{k}\right)\left(\genfrac{}{}{0ex}{}{n}{\ell }\right)\left(\genfrac{}{}{0ex}{}{n+\ell }{\ell }\right)E_k(x|\lambda ).$$

We end by noting that if we substitute $\lambda =0$ in any of our corollaries, then we get the well-known value of the polynomial $p(x)$. For instance, by setting $\lambda =0$, the last corollary gives that $y_n(x)={\sum }_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\frac{x^k}{2^k}$, as expected.