Identities and congruences involving the Fubini polynomials

TL;DR

This paper explores the umbral representation of Fubini polynomials to derive properties and congruences, including a key result involving prime numbers and polynomial congruences.

Contribution

It introduces new congruences involving Fubini polynomials and their umbral representations, extending understanding of their algebraic properties.

Findings

Proves that (f(F_x))^p ≡ f(F_x) mod p for prime p.

Derives several novel congruences involving Fubini polynomials.

Provides insights into the algebraic structure of Fubini polynomials.

Abstract

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $(f(F_{x}))^{p}\equiv f(F_{x})$ and we give other curious congruences.

Equations

$${\mathcal{F}}_n\left(x,r,s\right)=\underset{k=0}{\sum ^n}{\left\{\genfrac{}{}{0ex}{}{n+r}{k+r}\right\}}_r\left(k+s\right)!x^k.$$

$${\mathcal{F}}_0\left(x,r,s\right)$$

$${\mathcal{F}}_1\left(x,r,s\right)$$

$${\mathcal{F}}_2\left(x,r,s\right)$$

$${\left(x\right)}_n=\underset{k=0}{\sum ^n}{\left(-1\right)}^{n-k}{\left[\genfrac{}{}{0ex}{}{n+r}{k+r}\right]}_r{\left(x+r\right)}^k\text{ and }{\left(x+r\right)}^n=\underset{k=0}{\sum ^n}{\left\{\genfrac{}{}{0ex}{}{n+r}{k+r}\right\}}_r{\left(x\right)}_k,$$

$$\left(x+1\right)f\left({\mathbf{F}}_{\mathbf{x}}+r\right)=xf\left({\mathbf{F}}_{\mathbf{x}}+r+1\right)+f\left(r\right),$$

$${\left({\mathbf{F}}_{\mathbf{x}}+r\right)}_{n+r}=\left(n+r\right)!x^n{\left(x+1\right)}^r$$

$${\left({\mathbf{F}}_{\mathbf{x}}+r-s\right)}^n{\left({\mathbf{F}}_{\mathbf{x}}\right)}_s=x^s{\mathcal{F}}_n\left(x;r,s\right),$$

$${\left({\mathbf{F}}_{\mathbf{x}}+r\right)}^n{\left({\mathbf{F}}_{\mathbf{x}}+s\right)}_s={\left(x+1\right)}^s{\mathcal{F}}_n\left(x;r,s\right).$$

$$h_n(r):=\left(x+1\right){\left({\mathbf{F}}_{\mathbf{x}}+r\right)}^n-x{\left({\mathbf{F}}_{\mathbf{x}}+r+1\right)}^n$$

$${\left({\mathbf{F}}_{\mathbf{x}}+r\right)}_{n+r}=\underset{j=0}{\sum ^{n+r}}\left(\genfrac{}{}{0ex}{}{n+r}{j}\right){\left(r\right)}_j{\left({\mathbf{F}}_{\mathbf{x}}\right)}_{n+r-j}=\left(n+r\right)!x^n{\left(x+1\right)}^r.$$

$$\underset{j=0}{\sum ^n}{\left\{\genfrac{}{}{0ex}{}{n+r}{j+r}\right\}}_r{\left({\mathbf{F}}_{\mathbf{x}}\right)}_{j+s}$$

$$\underset{j=0}{\sum ^n}{\left\{\genfrac{}{}{0ex}{}{n+r}{j+r}\right\}}_r{\left({\mathbf{F}}_{\mathbf{x}}+s\right)}_{j+s}$$

$${\left(x+1\right)}^{s}f\left({\mathbf{F}}_{\mathbf{x}}+r-s\right){\left({\mathbf{F}}_{\mathbf{x}}\right)}_s=x^{s}f\left({\mathbf{F}}_{\mathbf{x}}+r\right){\left({\mathbf{F}}_{\mathbf{x}}+s\right)}_s.$$

$${\mathcal{P}}_n\left(x;r\right)=\underset{j=0}{\sum ^n}{\left(-1\right)}^j\left(\genfrac{}{}{0ex}{}{j+r}{r}\right)x^{n-j}\text{ and }{\mathcal{T}}_n\left(x;r\right)=\underset{j=0}{\sum ^n}\left(\genfrac{}{}{0ex}{}{n+r}{j+r}\right)x^j.$$

$${\left({\mathbf{F}}_{\mathbf{x}}-r-1\right)}_n$$

$${\left({\mathbf{F}}_{\mathbf{x}}+n+r\right)}_n$$

$${\left(x+1\right)}^{m}f\left({\mathbf{F}}_{\mathbf{x}}\right)-x^{m}f\left({\mathbf{F}}_{\mathbf{x}}+m\right)=\underset{k=0}{\sum ^{m-1}}f\left(k\right){\left(x+1\right)}^{m-1-k}{x}^k,m\ge 1.$$

$$\left(x+1\right)f\left({\mathbf{F}}_{\mathbf{x}}\right)-xf\left({\mathbf{F}}_{\mathbf{x}}+1\right)=f\left(0\right)+\sum _{k\ge 1}{a}_k\left(\left(x+1\right){\mathbf{F}}_{\mathbf{x}}^k-x{\left({\mathbf{F}}_{\mathbf{x}}+1\right)}^k\right)=f\left(0\right).$$

$${\left(x+1\right)}^{m+1}f\left({\mathbf{F}}_{\mathbf{x}}\right)$$

$$=\underset{k=0}{\sum ^{m-1}}{\left(x+1\right)}^{m-k}{x}^{k}f\left(k\right)+x^m\left(\left(x+1\right)f\left({\mathbf{F}}_{\mathbf{x}}+m\right)\right)$$

$${\left(x+1\right)}^{m+1}f\left({\mathbf{F}}_{\mathbf{x}}\right)$$

$$=\underset{k=0}{\sum ^{m-1}}{\left(x+1\right)}^{m-k}{x}^{k}f\left(k\right)+x^{m}f\left(m\right)+x^{m+1}f\left({\mathbf{F}}_{\mathbf{x}}+m+1\right)$$

$$=\underset{k=0}{\sum ^m}{\left(x+1\right)}^{m-k}{x}^{k}f\left(k\right)+x^{m+1}f\left({\mathbf{F}}_{\mathbf{x}}+m+1\right)$$

$$f({\mathbf{F}}_{\mathbf{x}})=\frac{1}{1+x}\sum _{k\ge 0}f(k){\left(\frac{x}{1+x}\right)}^k,x>-\frac{1}{2}.$$

$$RHS$$

$$=\underset{n\to \infty }{\lim }\left(f({\mathbf{F}}_{\mathbf{x}})-{\left(\frac{x}{1+x}\right)}^{n+1}f({\mathbf{F}}_{\mathbf{x}}+n+1)\right)=f({\mathbf{F}}_{\mathbf{x}})$$

$${\mathcal{F}}_n\left(x;r,s\right)=\frac{s!}{{\left(1+x\right)}^{s+1}}\sum _{k\ge 0}\left(\genfrac{}{}{0ex}{}{k+s}{s}\right)(k+r{)}^n{\left(\frac{x}{1+x}\right)}^k,x>-\frac{1}{2}.$$

$$x^r{\left(x+1\right)}^s{\mathcal{F}}_{n+1}\left(x;r,s+r\right)=x\frac{d}{dx}\left(x^r{\left(x+1\right)}^{s+1}{\mathcal{F}}_n\left(x;r,s+r\right)\right)$$

$$\left(1+x\right){\mathcal{F}}_{n-1}\left(x\right)=\underset{k=1}{\sum ^n}\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}\left(k-1\right)!x^k.$$

$$\left(1+x\right){\mathcal{F}}_{n-1}\left(x\right)$$

$$\log \left(1+\sum _{n\ge 1}\frac{{\mathcal{F}}_n\left(x;r,s\right)}{s!}\frac{t^n}{n!}\right)=\left(r+\left(s+1\right)x\right)t+\left(s+1\right)\left(x+1\right)\sum _{n\ge 2}{\mathcal{F}}_{n-1}\left(x\right)\frac{t^n}{n!},$$

$$\log \left(1+\sum _{n\ge 1}{\mathcal{F}}_n\left(x\right)\frac{t^n}{n!}\right)=xt+\left(x+1\right)\sum _{n\ge 2}{\mathcal{F}}_{n-1}\left(x\right)\frac{t^n}{n!}.$$

$$LHS$$

$$\underset{k=0}{\sum ^{p-1}}f\left(k\right){\left(x+1\right)}^{p-1-k}{x}^k\equiv f\left({\mathbf{F}}_{\mathbf{x}}\right).$$

$$\underset{k=0}{\sum ^{p-1}}{\left(r-s+k\right)}^n{\left(k\right)}_s{\left(x+1\right)}^{p-1-k}{x}^k$$

Full text

Identities and congruences involving the Fubini polynomials

Miloud Mihoubi11footnotemark: 1  and  Said Taharbouchet22footnotemark: 2

USTHB, Faculty of Mathematics, RECITS Laboratory, PB 32 El Alia 16111 Algiers, Algeria.

11footnotemark: 1

mmihoubi@usthb.dz  22footnotemark: 2said.taharbouchet@gmail.com

**Abstract. **In this paper, we investigate the umbral representation of the Fubini polynomials $\mathbf{F}_{\mathbf{x}}^{n}:=\mathcal{F}_{n}\left(x\right)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we show $\left(f\left(\mathbf{F}_{\mathbf{x}}\right)\right)^{p}\equiv f\left(\mathbf{F}_{\mathbf{x}}\right)$ and we give other curious congruences.

Keywords. Fubini umbra, Fubini polynomials, identities, congruences.

2000 MSC: 05A18, 05A40, 11A07.

1 Introduction

The Fubini numbers are quantities arising from enumerative combinatorics and have nice number-theoretic properties. In combinatorics, the $n$-th Fubini number $\mathcal{F}_{n}$ (named also the $n$-th ordered Bell number) counts the number of ways to partition the set $\left[n\right]:=\left\{1,\ldots,n\right\}$ into ordered subsets [2, 10]. The Fubini polynomials are defined by $\mathcal{F}_{n}\left(x\right)=\underset{k=0}{\overset{n}{\sum}}{n\brace k}k!x^{k}$ and satisfy the recurrence relation $\left(x+1\right)\mathcal{F}_{n}\left(x\right)=x\underset{j=0}{\overset{n}{\sum}}\binom{n}{j}\mathcal{F}_{j}\left(x\right),\ n\geq 1,$ where ${n\brace k}$ is the $\left(n,k\right)$-th Stirling number of the second kind [2, 18]. For $x=1$ we obtain the Fubini numbers $\mathcal{F}_{n}=\underset{k=0}{\overset{n}{\sum}}{n\brace k}k!$ [4, 6, 7, 9, 20, 10, 21].

More generally, let $\mathcal{F}_{n}\left(x,r,s\right)$ be the $n$-th $\left(r,s\right)$-Fubini polynomial defined by

[TABLE]

This polynomial generalizes the Fubini polynomial $\mathcal{F}_{n}\left(x\right)=\mathcal{F}_{n}\left(x;0,0\right)$ and the $r$-Fubini polynomial $\mathcal{F}_{n,r}\left(x\right)=\mathcal{F}_{n}\left(x;r,r\right)$ introduced by Mező [11]. Here, ${n\brace k}_{r}$ denotes the $\left(n,k\right)$-th $r$-Stirling number of the second kind [3]. One can see easily that

[TABLE]

As it shown below, these polynomials are also linked to the absolute $r$-Stirling numbers of first kind denoted by ${n\brack k}_{r}.$ Recall that the $r$-Stirling numbers can be defined by [3, 18]

[TABLE]

where $\left(\alpha\right)_{n}=\alpha\cdots\left(\alpha-n+1\right)$ if $n\geq 1$, $\left(\alpha\right)_{0}=1,$ ${n\brack k}={n\brack k}_{0}$ and ${n\brace k}={n\brace k}_{0},$

This work is motivated by application of the umbral calculus method to determine identities and congruences involving Bell numbers and polynomials in the works of Gessel [8], Sun et al. [19] and Benyattou et al. [1]. In this paper, we will talk about identities and congruences involving the $\left(r,s\right)$-Fubini polynomials based on the Fubini umbra defined by $\mathbf{F}_{\mathbf{x}}^{n}:=\mathcal{F}_{n}\left(x\right).$

For more information about umbral calculus, see [5, 8, 14, 15, 16, 17].

2 Identities for the $\left(r,s\right)$-Fubini polynomials

By definition of the Fubini umbra, it follows that the above recurrence relation can be rewritten as $\left(x+1\right)\mathbf{F}_{\mathbf{x}}^{n}=x\left(\mathbf{F}_{\mathbf{x}}+1\right)^{n},\ n\geq 1.$ Furthermore, we have

Proposition 1

Let $f$ be a polynomial and $r,s$ be non-negative integers. Then

[TABLE]

Proof. It suffices to show the first identity for $f\left(x\right)=x^{n}$. For r=0\we have $\left(x+1\right)\mathbf{F}_{\mathbf{x}}^{n}-x\left(\mathbf{F}_{\mathbf{x}}+1\right)^{n}=\delta_{\left(n=0\right)}.$ Assume it is true for $r-1,$ then if we set

[TABLE]

we obtain $h_{n}(r)=\underset{j=0}{\overset{n}{\sum}}\binom{n}{j}h_{j}(r-1)=\underset{j=0}{\overset{n}{\sum}}\binom{n}{j}\left(r-1\right)^{j}=r^{n},$ which concludes the induction step. For the other identities, since $\left(x\right)_{n}=\underset{k=0}{\overset{n}{\sum}}\left(-1\right)^{n-k}{n\brack k}x^{k}$ and $\left(x\right)_{n}$ is a sequence of binomial type [15, 12], we  obtain

[TABLE]

So, the polynomials $x^{s}\mathcal{F}_{n}\left(x;r,s\right)$ and $\left(x+1\right)^{s}\mathcal{F}_{n}\left(x,r,s\right)$ must be, respectively,

[TABLE]

$\Box$

The the last two identities of Proposition 1 lead to

Corollary 2

For any polynomial $f$ and any non-negative integers $r,s$ we have

[TABLE]

Proposition 3

Let $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ be the polynomials

[TABLE]

Then    $\left(\mathbf{F}_{\mathbf{x}}-r-1\right)_{n}=n!\mathcal{P}_{n}\left(x;r\right)$    and    $\left(\mathbf{F}_{\mathbf{x}}+n+r\right)_{n}=n!\mathcal{T}_{n}\left(x;r\right).$

Proof. It suffices to observe that

[TABLE]

$\Box$

The identities of the following two theorems depend on the choice of a polynomial $f$ and can be served to derive several identities and congruences for the $\left(r,s\right)$-Fubini polynomials.

Theorem 4

Let $f$ be a polynomial and let $m,s$ be non-negative integers. Then

[TABLE]

Proof. Set $f\left(x\right)=\underset{k\geq 0}{\sum}a_{k}x^{k}$ and use the first identity of Proposition 1 to obtain

[TABLE]

So, the identity is true for $m=1.$ Assume it is true for $m.$ Then

[TABLE]

and since $\left(x+1\right)f\left(\mathbf{F}_{\mathbf{x}}+m\right)-xf\left(\mathbf{F}_{\mathbf{x}}+m+1\right)=f\left(m\right),$ we can write

[TABLE]

which concludes the induction step.  $\Box$

Corollary 5

For any polynomial $f$ there holds

[TABLE]

Proof. For $m=1$ in Theorem 4 we get when we replace $f(x)$ by $f(x+r)$:

$f\left(r\right)=\left(x+1\right)f\left(\mathbf{F}_{\mathbf{x}}+r\right)-xf\left(\mathbf{F}_{\mathbf{x}}+r+1\right).$ Then

[TABLE]

which completes the proof.  $\Box$

Corollary 6

*Let $n,r,s$ be non-negative integers.

For $f\left(x\right)=\left(x+r\right)^{n}\left(x+s\right)_{s}$ or $\left(x+r-s\right)^{n}\left(x\right)_{s}$ in Corollary 5 we obtain*

[TABLE]

Corollary 7

For any integers $r\geq 0,\ s\geq 0$ and $n\geq 1$ the polynomial $\mathcal{F}_{n}\left(x,r,s+r\right)$ has only real non-positive roots.

Proof. From Corollary 6 we may state

[TABLE]

and using the definition and the recurrence relation of $r$-Stirling numbers we conclude that this property remains true for all real number $x.$ So, one can verify by induction on $n$ that the polynomial $\mathcal{F}_{n}\left(x;r,s+r\right),\ n\geq 1,$ has only real non-positive roots.    $\Box$

Lemma 8

For any non-negative integers $n\geq 2$ there holds

[TABLE]

Proof. From the definition of the Fubini polynomials, we have

[TABLE]

$\Box$

Proposition 9

Let $n,r,s$ be non-negative integers. Then

[TABLE]

In particular, for $r=s=0$ we get

[TABLE]

Proof. One can verify easily that the exponential generating function of the polynomials $\mathcal{F}_{n}\left(x;r,s\right)$ is to be  $s!\exp\left(rt\right)\left(1-x\left(\exp\left(t\right)-1\right)\right)^{-s-1}.$ Then, upon using this generating function and the last Lemma, we can write

[TABLE]

$\Box$

3 Congruences on the (r,s)-Fubini polynomials

In this section, we give some congruences involving the $(r,s)$-Fubini polynomials. Let $\mathbb{Z}_{p}$ be the ring of $p$-adic integers and for two polynomials $f\left(x\right),\ g\left(x\right)\in\mathbb{Z}_{p}\left[x\right],$ the congruence $f\left(x\right)\equiv g\left(x\right)\ \left(\mathop{\mathrm{m}od}p\mathbb{Z}_{p}\left[x\right]\right)$ means that the corresponding coefficients of $f\left(x\right)$ and $g\left(x\right)$ are congruent modulo $p.$ This congruence will be used later as $f\left(x\right)\equiv g\left(x\right)$ and we will use $a\equiv b$ instead $a\equiv b\ \left(\mathop{\mathrm{m}od}p\right)$.

Proposition 10

Let $n,r,s$ be non-negative integers and $p$ be a prime number. Then, for any polynomial $f$ with integer coefficients there holds

[TABLE]

In particular, for $f\left(x\right)=\left(x+r-s\right)^{n}\left(x\right)_{s}$ or $\left(x+r\right)^{n}\left(x+s\right)_{s}$ we get, respectively,

[TABLE]

Proof. For $m=p$ be a prime number, Theorem 4 implies

[TABLE]

For the particular cases, use Proposition 1.  $\Box$

Corollary 11

Let $n,r,s,m,q$ be non-negative integers and $p$ be a prime number. Then, for any polynomials $f$ and $g$ with integer coefficients there holds

[TABLE]

In particular, we have $\mathcal{F}_{mp+q}\left(x;r,s\right)\equiv\mathcal{F}_{m+q}\left(x;r,s\right).$

Proof. By Fermat’s little theorem and by twice application of Proposition 10 we may state

[TABLE]

which equals to the RHS.  $\Box$

Corollary 12

For any non-negative integers $m\geq 1,n,r,s$ and any prime number $p$, there hold

[TABLE]

where $r^{\prime}\equiv r$ and $r^{\prime}\in\left\{0,1,\ldots,p-1\right\}.$

Proof. Set $n=m\left(p-1\right)$ in the second particular case of Proposition 10.

If $r^{\prime}\neq 0$ we get

[TABLE]

and if $r^{\prime}=0$ we get

[TABLE]

which complete the proof.  $\Box$

Now, we give some curious congruences on $(r,s)$-Fubini polynomials and on $(r_{1},\ldots,r_{q})$-Fubini polynomials defined below.

Theorem 13

For any integers $n,m,r,s\geq 0$ and any prime number $p\nmid m,$ there holds

[TABLE]

Proof. Upon using the identity $x^{s}\mathcal{F}_{n}\left(x;r,s\right)=\left(\mathbf{F}_{\mathbf{x}}+r-s\right)^{n}\left(\mathbf{F}_{\mathbf{x}}\right)_{s}$ and the known congruence $\left(-m\right)^{-k}\equiv\binom{p-1}{k}m^{p-1-k}$ we obtain

[TABLE]

where $\delta$ is the Kronecker’s symbol, i.e. $\delta_{\left(n\geq 1\right)}=1$ if $n\geq 1$ and [math] otherwise.  $\Box$

Let $\mathbf{r}_{q}=\left(r_{1},\ldots,r_{q}\right)$ be a vector of non-negative integers and let

[TABLE]

where ${n+\left|\mathbf{r}_{q}\right|\brace j+r_{q}}_{\mathbf{r}_{q}}$ are the $\left(r_{1},\ldots,r_{q}\right)$-Stirling numbers defined by Mihoubi et al. [13]. This polynomial is a generalization of the $r$-Fubini polynomials $F_{n}\left(x;r\right):=\mathcal{F}_{n}\left(x;r,r\right)$.

Proposition 14

For any non-negative integers $n,m$ and any prime $p\nmid m,$ there holds

[TABLE]

In particular, for $q=1$ and $r_{q}=r$ we obtain

[TABLE]

Proof. By the identity $\left(\mathbf{F}_{\mathbf{x}}\right)_{n}=n!x^{n}$ and by [13, Th. 10] we have

[TABLE]

where $\underset{k=0}{\overset{\left|\mathbf{r}_{q}\right|}{\sum}}a_{k}\left(\mathbf{r}_{q}\right)u^{k}=\left(u\right)_{r_{1}}\cdots\left(u\right)_{r_{q}}.$ So, by application of Theorem 13 we get

[TABLE]

$\Box$

Remark 15

Since $x^{r_{q}}F_{n}\left(x;\mathbf{r}_{q}\right)=\mathbf{F}_{\mathbf{x}}^{n}\left(\mathbf{F}_{\mathbf{x}}\right)_{r_{1}}\cdots\left(\mathbf{F}_{\mathbf{x}}\right)_{r_{q}},$ then, for $f\left(x\right)=x^{mp+q}\left(x\right)_{r_{1}}\cdots\left(x\right)_{r_{q}}$ and $g\left(x\right)=1$ in Corollary 11 we obtain

[TABLE]

and for $f\left(x\right)=x^{m\left(p-1\right)}\left(x\right)_{r_{1}}\cdots\left(x\right)_{r_{q}}$ we get

[TABLE]

Corollary 16

Let $a_{0}\left(x\right),\ldots,a_{t}\left(x\right)$ be polynomials with integer coefficients,

[TABLE]

Then, for any non-negative integers $n,m,r,s$ and any prime $p\nmid m,$ there hold

[TABLE]

Proof. Theorem 13 implies

[TABLE]

$\Box$

4 Congruences involving $\mathcal{F}_{n}\left(x;r,s\right),$ $\mathcal{P}_{n}\left(x,r\right)$ and $\mathcal{T}_{n}\left(x,r\right)$

The following theorem gives connection in congruences between the polynomials $\mathcal{F}_{n}$ and $\mathcal{P}_{n}.$

Theorem 17

*Let $n,r$ be non-negative integers and $p$ be a prime number.

Then, for $m\in\left\{0,\ldots,p-1\right\}$ there holds*

[TABLE]

In particular, for $m=0,$ we get

[TABLE]

Proof. For $k<m$ we get $\left\langle m+1\right\rangle_{p-1-k}=0$ and for $m\leq k\leq p-1$ we have

[TABLE]

where $\left\langle x\right\rangle_{n}=x\left(x+1\right)\cdots\left(x+n-1\right)$ if $n\geq 1$ and $\left\langle x\right\rangle_{0}=1.$ Then

[TABLE]

But for $j\geq 1$ we have

[TABLE]

hence, it follows $LHS\equiv\left(-1\right)^{m}m!\left(r+m\right)^{n}\mathcal{P}_{p-1}\left(x,m\right)$.  $\Box$

The following theorem gives connection in congruences between the polynomials $\mathcal{F}_{n}$ and $\mathcal{T}_{n}$.

Theorem 18

For any integers $n,m,r\geq 0$ and any prime $p,$ there holds

[TABLE]

Proof. Upon using the identity $\left(x+1\right)^{s}\mathcal{F}_{n}\left(x;r,s\right)=\left(\mathbf{F}_{\mathbf{x}}+r\right)^{n}\left(\mathbf{F}_{\mathbf{x}}+s\right)_{s}$ and the known congruence $\left(m\right)_{p-1-k}\equiv\binom{p-1}{k}\left\langle-m\right\rangle_{p-1-k}$ we obtain

[TABLE]

$\Box$

The following theorem gives connection in congruences between the polynomials $\mathcal{T}_{n}$ and $\mathcal{P}_{n}.$

Theorem 19

For any integers $n,m,r\geq 0$ and any prime $p,$ there holds

[TABLE]

Proof. Upon using the congruence $\left(m\right)_{p-1-k}\equiv\binom{p-1}{k}\left\langle-m\right\rangle_{p-1-k}$ and Proposition 3 we obtain

[TABLE]

$\Box$

Corollary 20

Let $\mathcal{R}_{n,t}\left(x;r,s\right)$ be as in Corollary 16. Then, for any non-negative integers $n,m,r,s$ and any prime $p\nmid m,$ there holds

[TABLE]

Proof. Theorem 17 implies

[TABLE]

$\Box$