1. Introduction
Let p be a fixed prime number. Throughout this paper, Zp,Qp, and Cp denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively.

Let N∪{0}. Let UD(Zp) be space of uniformly differentiable functions on Zp. For f∈UD(Zp), the p-adic invariant integral on Zp is defined by
(1.1)∫Zpf(x)dμ(x)=limN→∞1pN∑x=0pN-1f(x),
see [1, 2].

From (1.1), we have
(1.2)∫Zpf(x+n)dμ(x)-∫Zpf(x)dμ(x)=∑l=0nf′(l), n∈N,
where f'(l)=(df(x)/dx)∣x=l (see [1–6]). Let F be the set of all formal power series in the variable t over Cp with
(1.3)F={f(t)=∑k=0∞akk!tk∣ak∈Cp}.
Let ℙ=Cp[x] and let ℙ* denote the vector space of all linear functional on ℙ.

The formal power series,
(1.4)f(t)=∑k=0∞akk!tk∈F,
defines a linear functional on ℙ by setting
(1.5)〈f(t)∣xn〉=an, ∀n≥0,
see [7, 8].

In particular, by (1.4) and (1.5), we get
(1.6)〈tk∣xn〉=n!δn,k,
where δn,k is the Kronecker symbol (see [7]). Here, F denotes both the algebra of formal power series in t and the vector space of all linear functional on ℙ, so an element f(t) of F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra. The umbral calculus is the study of umbral algebra.

The order o(f(t)) of power series f(t)(≠0) is the smallest integer k for which ak does not vanish. We define o(f(t))=∞ if f(t)=0. From the definition of order, we note that o(f(t)g(t))=o(f(t))+o(g(t)) and o(f(t)+g(t))≥min{o(f(t)),o(g(t))}.

The series f(t) has a multiplicative inverse, denoted by f(t)-1 or 1/f(t), if and only if o(f(t))=0.

Such a series is called invertible series. A series f(t) for which o(f(t))=1 is called a delta series (see [7, 8]). Let f(t),g(t)∈F. Then, we have
(1.7)〈f(t)g(t)∣p(x)〉=〈f(t)∣g(t)p(x)〉=〈g(t)∣f(t)p(x)〉.
By (1.5) and (1.6), we get
(1.8)〈eyt∣xn〉=yn, 〈eyt∣p(x)〉=p(y),
see [7].

Notice that for all f(t) in F,
(1.9)f(t)=∑k=0∞〈f(t)∣xk〉k!tk,
and for all polynomials p(x),
(1.10)p(x)=∑k≥0〈tk∣p(x)〉k!xk,
see [7, 8].

Let f1(t),f2(t),…,fm(t)∈F. Then, we have
(1.11)〈f1(t)f2(t)⋯fm(t)∣xn〉=∑(ni1,…,im)〈f1(t)∣xi1〉⋯〈fm(t)∣xim〉,
where the sum is over all nonnegative integers i1,i2,…,im such that i1+⋯+im=n (see [8]).

By (1.10), we get
(1.12)p(k)(x)=dkp(x)dxk=∑l=kn〈tl∣p(x)〉l!l(l-1)⋯(l-k+1)xl-k.
Thus, from (1.12), we have
(1.13)p(k)(0)=〈tk∣p(x)〉=〈1∣p(k)(x)〉,
see [7].

By (1.13), we get
(1.14)tkp(x)=p(k)(x)=dk(p(x))dxk.
Thus, by (1.14), we see that
(1.15)eytp(x)=p(x+y).
Let us assume that sn(x) is a polynomial of degree n. Suppose that f(t),g(t)∈F with o(f(t))=1 and o(g(t))=0. Then, there exists a unique sequence sn(x) of polynomials satisfying 〈g(t)f(t)k∣sn(x)〉=n!δn,k for all n,k≥0.

The sequence sn(x) is called the Sheffer sequence for (g(t),f(t)), which is denoted by sn(x)~(g(t),f(t)).

The Sheffer sequence for (g(t),t) is called the Appell sequence for g(t), or sn(x) is Appell for g(t), which is indicated by sn(x)~(g(t),t).

For p(x)∈ℙ, it is known that
(1.16)〈f(t)∣xp(x)〉=〈∂tf(t)∣p(x)〉=〈f′(t)∣p(x)〉,〈eyt-1∣p(x)〉=p(y)-p(0),
see [7, 8].

Let sn(x)~(g(t),f(t)). Then, we have
(1.17)h(t)=∑k=0∞〈h(t)∣sk(x)〉k!g(t)f(t)k, h(t)∈F,(1.18)p(x)=∑k=0∞〈g(t)f(t)k∣p(x)〉k!sk(x), p(x)∈ℙ,(1.19)1g(f-(t))eyf-(t)=∑k=0∞sk(y)k!tk, for any y∈Cp,
where f-(t) is the compositional inverse of f(t), and
(1.20)f(t)sn(x)=nsn-1(x),
see [7, 8].

We recall that the Bernoulli polynomials are defined by the generating function to be
(1.21)tet-1ext=eB(x)t=∑n=0∞Bn(x)tnn!,
with the usual convention about replacing Bn(x) by Bn(x) (see [1–16]).

In the special case, x=0,Bn(0)=Bn are called the nth Bernoulli numbers. By (1.21), we easily get
(1.22)Bn(x)=(B+x)n=∑l=0n(nl)Blxn-l=∑l=0n(nl)Bn-lxl.
Thus, by (1.22), we see that Bn(x) is a monic polynomial of degree n. It is easy to show that
(1.23)B0=1, Bn(1)-Bn=δ1,n,
see [13–15].

From (1.2), we can derive the following equation:
(1.24)∫Zpf(x+1)dμ(x)-∫Zpf(x)dμ(x)=f′(0).
Let us take f(x)=etx∈UD(Zp). Then, from (1.21), (1.22), (1.23), and (1.24), we have
(1.25)∫Zpxndμ(x)=Bn, ∫Zp(x+y)ndμ(y)=Bn(x),
where n≥0 (see [1, 2]). Recently, Dere and simsek have studied applications of umbral algebra to some special functions (see [7]). In this paper, we investigate some properties of umbral calculus associated with p-adic invariant integrals on Zp. From our properties, we can derive some interesting identities of Bernoulli polynomials.

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Let sn(x) be an Appell sequence for g(t). By (1.19), we get
(2.1)1g(t)xn=sn(x), iff xn=g(t)sn(x).
Let us take g(t)=((et-1)/t)∈F. Then, g(t) is clearly invertible series. From (1.21) and (2.1), we have
(2.2)∑k=0∞Bk(x)k!tk=1g(t)ext.
Thus, by (2.2), we get
(2.3)1g(t)xn=Bn(x), tBn(x)=Bn′(x)=nBn-1(x), (n≥0).
From (1.21), (2.1), and (2.3), we note that Bn(x) is an Appell sequence for g(t)=(et-1)/t.

Let us take the derivative with respect to t on both sides of (2.2). Then, we have
(2.4)∑k=1∞Bk(x)k!ktk-1=xg(t)ext-extg′(t)g(t)2=∑k=0∞{xxkg(t)-xkg(t)g′(t)g(t)}tkk!.
Thus, by (2.4), we get
(2.5)Bk+1(x)=xxkg(t)-xkg(t)g′(t)g(t)=(x-g′(t)g(t) )Bk(x),
where k≥0. (2.6)∫Zpe(x+y+1)tdμ(y)-∫Zpe(x+y)tdμ(y)=text.
Thus, by (2.6), we get
(2.7)∫Zp(x+y+1)ndμ(y)-∫Zp(x+y)nμ(y)=nxn-1, (n≥0).
From (1.25) and (2.7), we have
(2.8)Bn(x+1)-Bn(x)=nxn-1, (n≥0).
By (2.5), we see that
(2.9)g(t)Bk+1(x)=g(t)xBk(x)-g′(t)Bk(x),
Thus, by (2.9), we have
(2.10)(et-1)Bk+1(x)=(et-1)xBk(x)-(et-g(t))Bk(x), (k≥0),
and we can derive the following equation.

From (2.3) and (2.10),
(2.11)Bk+1(x+1)-Bk+1(x)=(x+1)Bk(x+1)-xBk(x)-Bk(x+1)+xk, (k≥0).
By (2.8) and (2.11), we see that
(2.12)Bk+1(x+1)=Bk+1(x)+(k+1)xk.
Therefore, by (2.5), we obtain the following theorem.

Theorem 2.1.
For k∈Z+, one has
(2.13)Bk+1(x)=(x-g′(t)g(t))Bk,
where g′(t)=dg(t)/dt.

Corollary 2.2.
For ≥0, one has
(2.14)Bk+1(x+1)=Bk+1(x)+(k+1)xk.

Let us consider the linear functional f(t) that satisfies
(2.15)〈f(t)∣p(x)〉=∫Zpp(u)dμ(u),
for all polynomials p(x). It can be determined from (1.9) that
(2.16)f(t)=∑k=0∞〈f(t)∣xk〉k!tk=∑k=0∞∫Zpukdμ(u)tkk!=∫Zpeutdμ(u).
By (1.24) and (2.16), we get
(2.17)f(t)=∫Zpeutdμ(u)=tet-1.
Therefore, by (2.17), we obtain the following theorem.

Theorem 2.3.
For p(x)∈P, one has
(2.18)〈∫Zpeutdμ(u)∣p(x)〉=∫Zpp(u)dμ(u).
That is
(2.19)〈tet-1∣p(x)〉=∫Zpp(u)dμ(u).
In particular, one has
(2.20)Bn=〈∫Zpeutdμ(u)∣xn〉.

From (1.24), one has
(2.21)∑n=0∫Zp(x+y)ndμ(y)tnn!=∫Zpe(x+y)tdμ(y)=∑n=0∞∫Zpeytdμ(y)xntnn!.
By (1.25) and (2.21), we get
(2.22)Bn(x)=∫Zp(x+y)ndμ(y)=∫Zpeytdμ(y)xn,
where n≥0.

Therefore, by (2.22), we obtain the following theorem.

Theorem 2.4.
For p(x)∈ℙ, we have
(2.23)∫Zpp(x+y)dμ(y)=∫Zpeytdμ(y)p(x)=tet-1p(x).
In particular, one obtains
(2.24)Bn(x)=∫Zp(x+y)ndμ(y)=∫Zpeytdμ(y)xn=tet-1xn.

The higher order Bernoulli polynomials Bn(r)(x) are defined by
(2.25)∫Zp⋯∫Zpe(x1+x2+⋯+xr+x)tdμ(x1)⋯dμ(xr) =(tet-1)rext=∑n=0∞Bn(r)(x)tnn!.
In the special case, x=0, Bn(r)(0)=Bn(r) are called the nth Bernoulli numbers of order r (∈N). From (2.25), we note that
(2.26)∫Zp⋯∫Zp(x1+⋯+xr)ndμ(x1)⋯dμ(xr) =∑i1+⋯+ir=n(ni1,…,ir)∫Zpx1i1dμ(x1)∫Zpx2i2dμ(x2)⋯∫Zpxrirdμ(xr) =∑i1+⋯+ir=n(ni1,…,ir)Bi1⋯Bir=Bn(r).
By (2.25) and (2.26), we get
(2.27)Bn(r)(x)=∑l=0n(nl)Bn-l(r)xl.
From (2.26) and (2.27), we note that Bn(r)(x) is a monic polynomial of degree n with coefficients in Q. For r∈N, let us assume that
(2.28)g(r)(t)=(∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr))-1=(et-1t)r.
By (2.28), we easily see that g(r)(t) is an invertible series. From (2.25) and (2.28), we have
(2.29)extg(r)(t)=∫Zp⋯∫Zpe(x1+⋯+xr+x)tdμ(x1)⋯dμ(xr)=∑n=0∞Bn(r)(x)tnn!,tBn(r)(x)=nBn-1(r)(x).
From (2.29), we note that Bn(r) is an Appell sequence for g(r)(t). Therefore, by (2.29), we obtain the following theorem.

Theorem 2.5.
For p(x)∈ℙ and r∈N, one has
(2.30)∫Zp⋯∫Zpp(x1+⋯+xr+x)dμ(x1)⋯dμ(xr)=(tet-1)rp(x).
In particular, the Bernoulli polynomials of order r are given by
(2.31)Bn(r)(x)=(tet-1)rxn=∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr)xn.
That is
(2.32)Bn(r)(x)~((et-1t)r,t).

Let us consider the linear functional f(r)(t) that satisfies
(2.33)〈f(r)(t)∣p(x)〉=∫Zp⋯∫Zpp(x1+⋯+xr)dμ(x1)⋯dμ(xr),
for all polynomials p(x). It can be determined from (1.9) that
(2.34)f(r)(t)=∑k=0∞〈f(r)(t)∣xk〉k!tk=∑k=0∞∫Zp⋯∫Zp(x1+⋯+xr)kdμ(x1)⋯dμ(xr)tkk!=∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr)=(tet-1)r.
Therefore, by (2.34), we obtain the following theorem.

Theorem 2.6.
For p(x)∈ℙ, one has
(2.35)〈∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr)∣p(x)〉 =∫Zp⋯∫Zpp(x1+⋯+xr)dμ(x1)⋯dμ(xr).
That is
(2.36)〈(tet-1)r∣p(x)〉=∫Zp⋯∫Zpp(x1+⋯+xr)dμ(x1)⋯dμ(xr).
In particular, one gets
(2.37)Bn(r)=〈∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr)∣xn〉.

Remark 2.7.
From (1.11), we note that
(2.38)〈∫Zp⋯∫Zpe(x1+⋯+xr)tdμ(x1)⋯dμ(xr)∣xn〉 =∑n=i1+⋯+ir(ni1,…,ir)〈∫Zpex1tdμ(x1)∣xi1〉⋯〈∫Zpexrtdμ(xr)∣xir〉.
By Theorems 2.3 and 2.6 and (2.38), we get
(2.39)Bn(r)=∑n=i1+⋯+ir(ni1,…,ir)Bi1⋯Bir.
Let sn(x) be the Sheffer sequence for (g(t),f(t)).

Then the Sheffer identity is given by
(2.40)sn(x+y)=∑k=0n(nk)pk(y)sn-k(x),
see [7, 8], where pk(y)=g(t)sk(y). From Theorem 2.5 and (2.40), we have
(2.41)Bn(r)(x+y)=∑k=0n(nk)Bn-k(r)(x)xk.