Furthermore, for any prime numberpand any polynomialf with integer coefficients, we show.f .wx//pf .wx/(mod p) and we give other curious congruences

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 395–408 DOI: 10.18514/MMN.2019.2498

IDENTITIES AND CONGRUENCES INVOLVING THE GEOMETRIC POLYNOMIALS

MILOUD MIHOUBI AND SAID TAHARBOUCHET Received 13 January, 2018

Abstract. In this paper, we investigate the umbral representation of the geometric polynomials wⁿ_xWDwn.x/to derive some properties involving these polynomials. Furthermore, for any prime numberpand any polynomialf with integer coefficients, we show.f .wx//^pf .wx/(mod p) and we give other curious congruences.

2010Mathematics Subject Classification: 05A18; 05A40; 11A07

Keywords: geometric umbra, geometric polynomials, identities, congruences

1. INTRODUCTION

The geometric numbers are quantities arising from enumerative combinatorics and have nice number-theoretic properties. In combinatorics, the n-th geometric num- ber (named also the n-th ordered Bell number) counts the number of ways to par- tition the set ŒnWD f1; : : : ; nginto ordered subsets [2,3,6]. The geometric poly- nomials are defined bywn.x/DPn

kD0

˚n

k kŠx^k and satisfy the recurrence relation .xC1/wn.x/DxPn

jD0 n j

wj.x/; n1; [9], where ˚n

k is the .n; k/-th Stirling number of the second kind [2,26]. These polynomials have attracted attention from many researchers, see for instance [9,10,15–17]. ForxD1we obtain the geomet- ric numberswnWDwn.1/DPn

kD0

˚_n

k kŠ, for more information about these numbers, see [6–8,11,12,14,28,29]. More generally, letwn.xIr; s/be then-th.r; s/-geometric polynomial defined by

wn.xIr; s/D

n

X

kD0

(nCr kCr

)

r

.kCs/Šx^k:

This polynomial generalizes the geometric polynomialwn.x/Dwn.xI0; 0/and the polynomialwn.xIr; r/ introduced by Mez˝o [18]. Here, ˚n

k r denotes the.n; k/-th r-Stirling number of the second kind [4]. One can see easily that

w0.xIr; s/DsŠ;

w1.xIr; s/DsŠ.rC.sC1/x/;

c 2019 Miskolc University Press

w2.x; r; s/DsŠ.r²C.2rC1/.sC1/xC.sC1/.sC2/x²/:

We note that this generalization can be viewed as a particular case of that defined by Kargin et al. [16]. As it shown below, these polynomials are also linked to the absoluter-Stirling numbers of first kind denoted byn

k

r: Recall that ther-Stirling numbers can be defined by [4,26]

.x/nD

n

X

kD0

. 1/^{n k}

"

nCr kCr

#

r

.xCr/^k and.xCr/ⁿD

n

X

kD0

(nCr kCr

)

r

.x/_k;

where.˛/nD˛ .˛ nC1/ifn1,.˛/0D1.

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 [13], Sun et al. [27], Mez˝o et al. [19] and Benyattou et al. [1]. In this paper, we will talk about identities and congruences involving the.r; s/-geometric polynomials based on the geometric umbra defined bywⁿ_x WDwn.x/:For more information about umbral calculus, see [5,13,22–25].

2. IDENTITIES INVOLVING THE.r; s/-GEOMETRIC POLYNOMIALS

The above recurrence relation is equivalent to .xC1/wⁿ_x Dx.w_xC1/ⁿ; n1.

Furthermore, we have

Proposition 1. Letf be a polynomial andr; sbe non-negative integers. Then .xC1/f .w_xCr/Dxf .w_xCrC1/Cf .r/;

.w_xCr/nCrD.nCr/Šxⁿ.xC1/^r; .w_xCr s/ⁿ.w_x/sDx^swn.xIr; s/;

.w_xCr/ⁿ.w_xCs/sD.xC1/^swn.xIr; s/:

Proof. It suffices to show the first identity forf .x/Dxⁿ. For r D0 we have .xC1/wⁿ_x x.w_xC1/ⁿDı_.nD0/:Assume it is true forr 1;then if we set

hn.r/WD.xC1/.wxCr/ⁿ x.wxCrC1/ⁿ we obtainhn.r/DPn

jD0 n j

hj.r 1/DPn jD0

n j

.r 1/^j Drⁿ;which concludes the induction step. For the other identities, since.x/nDPn

kD0. 1/^{n k}_n

k

x^k and .x/nis a sequence of binomial type [20,23], we obtain

.w_xCr/nCrD

nCr

X

jD0

nCr j

!

.r/j.w_x/nCr j D.nCr/Šxⁿ.xC1/^r:

So, the polynomialsx^swn.xIr; s/and.xC1/^swn.x; r; s/must be, respectively,

n

X

jD0

(nCr jCr

)

r

.w_x/jCsD

n

X

jD0

(nCr jCr

)

r

.w_x s/j.w_x/sD.w_xCr s/ⁿ.w_x/s;

n

X

jD0

(nCr jCr

)

r

.w_xCs/jCsD

n

X

jD0

(nCr jCr

)

r

.w_x/j.w_xCs/sD.w_xCr/ⁿ.w_xCs/s:

The last two identities of Proposition1lead to:

Corollary 1. Let r; s be non-negative integers and f be a polynomial. Then .xC1/^sf .w_xCr s/.w_x/sDx^sf .w_xCr/.w_xCs/s:

Proposition 2. LetPnandTnbe the polynomials

Pn.xIr/D

n

X

jD0

. 1/^j jCr r

!

x^{n j} and Tn.xIr/D

n

X

jD0

nCr jCr

! x^j:

Then .w_x r 1/nDnŠPn.xIr/ and .w_xCnCr/nDnŠTn.xIr/:

Proof. It suffices to observe that

.w_x r 1/nD

n

X

jD0

n j

!

. r 1/j.w_x/n j DnŠ

n

X

jD0

. 1/^j jCr r

! x^{n j};

.w_xCnCr/nD

n

X

jD0

n j

!

.nCr/n j.w_x/j DnŠ

n

X

jD0

nCr jCr

! x^j:

The following theorem can be served to derive several identities and congruences for the.r; s/-geometric polynomials.

Theorem 1. Letm; sbe non-negative integers andf be a polynomial. Then .xC1/^mf .w_x/ x^mf .w_xCm/D

m 1

X

kD0

f .k/.xC1/^{m 1 k}x^k; m1:

Proof. Setf .x/DPn

kD0a_kx^k and use Proposition1to obtain .xC1/f .w_x/ xf .w_xC1/Df .0/C

n

X

kD0

a_k

.xC1/w^k_x x.w_xC1/^k

Df .0/:

So, the identity is true formD1:Assume it is true form:Then .xC1/^mC1f .w_x/D.xC1/

^{m 1} X

kD0

.xC1/^{m 1 k}x^kf .k/Cx^mf .w_xCm/

D

m 1

X

kD0

.xC1/^{m k}x^kf .k/Cx^m.xC1/f .w_xCm/

and since.xC1/f .w_xCm/ xf .w_xCmC1/Df .m/;we can write

.xC1/^mC1f .w_x/D

m 1

X

kD0

.xC1/^{m k}x^kf .k/Cx^m

xf .w_xCmC1/Cf .m/

D

m 1

X

kD0

.xC1/^{m k}x^kf .k/Cx^mf .m/Cx^mC1f .w_xCmC1/

D

m

X

kD0

.xC1/^{m k}x^kf .k/Cx^mC1f .w_xCmC1/

which concludes the induction step.

We note that forf .x/DxⁿandxD1 in Theorem 1 we obtain Proposition 3.3 given in [8].

Corollary 2. For any polynomialf there holds

f .w_x/D 1 1Cx

X

k0

f .k/

x 1Cx

k

; x > 1 2:

Proof. FormD1 in Theorem1, when we replacef .x/byf .xCr/ we get the identityf .r/D.xC1/f .w_xCr/ xf .w_xCrC1/. Then

RHSD lim

n!1

1 1Cx

n

X

kD0

x 1Cx

k

.xC1/f .w_xCk/ xf .w_xCkC1/

D lim

n!1.f .w_x/

x 1Cx

nC1

f .w_xCnC1//Df .w_x/

which completes the proof.

Corollary 3. Letn; r; sbe non-negative integers.

Forf .x/D.xCr/ⁿ.xCs/sor.xCr s/ⁿ.x/sin Corollary2we obtain

wn.xIr; s/D sŠ .1Cx/^sC1

X

k0

kCs s

!

.kCr/ⁿ x

1Cx k

; x > 1 2:

Corollary 4. For any integersr0; s0andn1the polynomialwn.x; r; sC r/has only real non-positive zeros.

Proof. From Corollary3we may state x^r.xC1/^swnC1.xIr; sCr/Dx d

dx

x^r.xC1/^sC1wn.xIr; sCr/

and using the recurrence relation ofr-Stirling numbers we conclude that this identity remains true for all real numberx. So, by induction onn, it follows thatwn.xIr; sC

r/; n1;has only real non-positive zeros.

Lemma 1. For any non-negative integersn2there holds

.1Cx/wn 1.x/D

n

X

kD1

(n k )

.k 1/Šx^k:

Proof. From the definition of geometric polynomials, we have

.1Cx/wn 1.x/D

n 1

X

kD1

(n 1 k

) kŠx^kC

n 1

X

kD1

(n 1 k

) kŠx^kC1

D

n

X

kD1

k

(n 1 k

) C

(n 1 k 1

)

.k 1/Šx^k

D

n

X

kD1

(n k

)

.k 1/Šx^k:

For more explicit formulae for geometric polynomials, see for example [15].

Proposition 3. Letn; r; sbe non-negative integers. Then log

1CX

n1

wn.xIr; s/

sŠ tⁿ nŠ

D.rC.sC1/x/tC.sC1/.xC1/X

n2

wn 1.x/tⁿ nŠ: In particular, forrDsD0we get

log

1CX

n1

wn.x/tⁿ nŠ

DxtC.xC1/X

n2

wn 1.x/tⁿ nŠ:

Proof. One can verify easily that the exponential generating function of the poly- nomialswn.xIr; s/is to be sŠexp.rt /.1 x.exp.t / 1// ^{s 1}:Then, upon using this generating function and the last Lemma, we can write

LHS Drt .sC1/ln.1 x.exp.t / 1//

DrtC.sC1/X

k1

x^k

k .exp.t / 1/^k

DrtC.sC1/X

k1

.k 1/Šx^kX

nk

(n k

)tⁿ nŠ

DrtC.sC1/xtC.sC1/X

n2

tⁿ nŠ

n

X

kD1

(n k )

.k 1/Šx^k

D.rC.sC1/x/tC.sC1/.xC1/X

n2

wn 1.x/tⁿ nŠ:

3. CONGRUENCES INVOLVING THE(R,S)-GEOMETRIC POLYNOMIALS

In this section, we give some congruences involving the.r; s/-geometric polyno- mials. LetZp be the ring ofp-adic integers and for two polynomialsf .x/; g.x/2 ZpŒx;the congruencef .x/g.x/ (modpZpŒx ) means that the corresponding coefficients off .x/andg.x/are congruent modulop:This congruence will be used later asf .x/g.x/and we will useabinsteadab(modp).

Proposition 4. Letn; r; sbe non-negative integers andpbe a prime number. Then, for any polynomialf with integer coefficients there holds

p 1

X

kD0

f .k/.xC1/^{p 1 k}x^kf .w_x/:

In particular, forf .x/D.xCr s/ⁿ.x/sor.xCr/ⁿ.xCs/swe get, respectively,

p 1

X

kD0

.r sCk/ⁿ.k/s.xC1/^{p 1 k}x^k x^swn.xIr; s/;

p 1

X

kD0

.rCk/ⁿ.sCk/s.xC1/^{p 1 k}x^k .xC1/^swn.xIr; s/:

Proof. FormDpbe a prime number, Theorem1implies

LHS D.xC1/^pf .w_x/ x^pf .w_xCp/.x^pC1/f .w_x/ x^pf .w_x/Df .w_x/:

For the particular cases, use Proposition1.

Corollary 5. Letn; r; s; m; qbe non-negative integers andp be a prime number.

Then, for any polynomialsf andgwith integer coefficients there holds .f .w_x//^pg.w_x/f .w_x/g.w_x/:

In particular, we havewmpCq.xIr; s/wmCq.xIr; s/:

Proof. By Fermat’s little theorem and by twice application of Proposition 4 we may state

LHS

p 1

X

kD0

.f .k//^pg.k/.xC1/^{p 1 k}x^k

p 1

X

kD0

f .k/g.k/.xC1/^{p 1 k}x^kDRHS:

We note that, forf .x/Dx^m; g.x/Dx^qandxD1, Corollary5may be seen as a particular case of Theorem 3.1 given in [8].

Corollary 6. For any non-negative integersm1; n; r; sand any prime number p, there hold

.xC1/^sC1.wm.p 1/.xIr; s/ sŠ/ .s r⁰/s.xC1/^r0x^{p r0}; r⁰¤0;

.xC1/^sC1.w_{m.p 1/}.xIr; s/ sŠ/ sŠ.x^pC1/; r⁰D0;

wherer⁰randr⁰2 f0; 1; : : : ; p 1g:

Proof. SetnDm.p 1/in Proposition4. Ifr⁰¤0we get

.xC1/^sw_{m.p 1/}.xIr; s/

p 1

X

kD0

.r⁰Ck/^{m.p 1/}.sCk/s.xC1/^{p 1 k}x^k

p 1

X

kD0; r⁰Ck¤p

.sCk/s.xC1/^{p 1 k}x^k

D

p 1

X

kD0

.sCk/s.xC1/^{p 1 k}x^k

.s r⁰Cp/s.xC1/^{r0 1}x^{p r0}

.xC1/^sw0.xI0; s/ .s r⁰/s.xC1/^{r0 1}x^{p r0} sŠ.xC1/^s .s r⁰/s.xC1/^{r0 1}x^{p r0}

and ifr⁰D0we get

.xC1/^sC1w_{m.p 1/}.xIr; s/

p 1

X

kD1

.sCk/s.xC1/^{p k}x^k

D

p 1

X

kD0

.sCk/s.xC1/^{p k}x^k sŠ.xC1/^p

D.xC1/^sC1w0.xI0; s/ sŠ.xC1/^p DsŠ.xC1/^sC1 sŠ.x^pC1/:

which complete the proof.

Remark1. ForrDsDm 1D0in Corollary6ornDpin Lemma1we obtain .xC1/wp 1.x/x x^p which gives forxD1the known congruencewp 10, see [8].

Now, we give some curious congruences on.r; s/-geometric polynomials and on .r1; : : : ; rq/-geometric polynomials defined below.

Theorem 2. For any integersn; m; r; s0and any prime numberp −m;there holds

p 1

X

kD1

w_nCk.xIr; s/

. m/^k . m/ⁿ.wp 1.xIrCm; s/ sŠ/:

Proof. Upon using the identityx^swn.xIr; s/D.w_xCr s/ⁿ.w_x/sand the known congruence. m/ ^{k p 1}_k

m^{p 1 k}we obtain

x^sLHS

p 1

X

kD0

p 1 k

!

m^{p 1 k}.w_xCr s/^nCk.w_x/s

D.w_xCr s/ⁿ.w_xCrCm s/^{p 1}.w_x/s

D

n

X

jD0

n j

!

. m/^{n j}.w_xCrCm s/^{jCp 1}.w_x/s

D. m/ⁿ.w_xCrCm s/^{p 1}.w_x/s

Cı_.n1/

n

X

jD1

n j

!

. m/^{n j}.w_xCrCm s/^{jCp 1}.w_x/s

Dx^s. m/ⁿwp 1.xIrCm; s/

Cı_.n1/x^s

n

X

jD1

n j

!

. m/^{n j}wpCj 1.xIrCm; s/

x^s. m/ⁿwp 1.xIrCm; s/

Cı_.n1/x^s

n

X

jD1

n j

!

. m/^{n j}wj.xIrCm; s/

Dx^s. m/ⁿw_{p 1}.xIrCm; s/Cı_.n1/x^s.w_n.xIr; s/ . m/ⁿsŠ/

Dx^sŒ. m/ⁿwp 1.xIrCm; s/Cwn.xIr; s/ . m/ⁿsŠ;

whereıis the Kronecker’s symbol, i.e.ı_.n1/D1ifn1and0otherwise.

LetrqD.r1; : : : ; rq/be a vector of non-negative integers and let

wn.xIrq/D

nCjrq 1j

X

jD0

(nC jrqj jCrq

)

rq

.jCrq/Šx^j; 0r1 rq;

where˚nCjrqj jCrq rq

are the.r1; : : : ; rq/-Stirling numbers defined by Mihoubi et al. [21].

This polynomial is a generalization of the r-geometric polynomials wn.xIr/ WD wn.xIr; r/.

Proposition 5. For any non-negative integersn; m and any primep−m; there holds

x^rq

p 1

X

kD1

w_nCk.xIrq/

. m/^k . m/ⁿ. m/r1 . m/rq.wp 1.xIm; 0/ 1/:

In particular, forqD1andrqDr we obtain

x^r

p 1

X

kD1

w_nCk.xIr; r/

. m/^k . m/ⁿ. m/r.wp 1.xIm; 0/ 1/:

Proof. By the identity.w_x/nDnŠxⁿand by [21, Th. 10] we have

x^rqwn.xIrq/D

nCjrq 1j

X

jD0

(nC jrqj jCrq

)

rq

.w_x/jCrq

D

nCjrq 1j

X

jD0

(nC jrqj jCrq

)

rq

.w_x rq/j.w_x/rq

Dwⁿ_x.w_x/r₁ .w_x/r_q

D

jrqj

X

kD0

a_k.r_q/wⁿ_x^Ck

D

jrqj

X

jD0

aj.rq/wnCj.x/;

wherePjrqj

kD0ak.rq/u^kD.u/r1 .u/rq:So, by application of Theorem2we get

x^rq

p 1

X

kD1

w_nCk.xIrq/ . m/^k D

jrqj

X

jD0

aj.rq/

p 1

X

kD1

w_nCjCk.xI0; 0/

. m/^k

jrqj

X

jD0

aj.rq/. m/^nCj.wp 1.xIm; 0/ 1/

D. m/ⁿ. m/r1 . m/rq.wp 1.xIm; 0/ 1/:

Remark2. Sincex^rqwn.xIrq/Dwⁿ_x.w_x/r1 .w_x/rq;then, for g.x/Dx^q.x/r1 .x/rq andf .x/Dx^min Corollary5we obtain

wmpCq.xIrq/wmCq.xIrq/;

w_{m.p 1/}.xIrq/w0.xIrq/; r1 rq¤0; m0:

Corollary 7. Leta0.x/; : : : ; at.x/be polynomials with integer coefficients,

Rn;t.xIr; s/D

t

X

iD0

ai.x/wnCi.xIr; s/ and Lt.x; y/D

t

X

iD0

ai.x/yⁱ:

Then, for any non-negative integersn; m; r; sand any primep−m;there hold

p 1

X

kD1

RnCk;t.xIr; s/

. m/^k . m/ⁿLt.x; m/.wp 1.xIrCm; s/ sŠ/:

Proof. Theorem2implies

p 1

X

kD1

R_nCk;t.xIr; s/

. m/^k D

t

X

jD0

aj.x/

p 1

X

kD1

w_nCkCj.xIr; s/

. m/^k

t

X

jD0

aj.x/. m/^nCj.wp 1.xIrCm; s/ sŠ/

D. m/ⁿLt.x; m/.wp 1.xIrCm; s/ sŠ/:

4. CONGRUENCES INVOLVINGwn.xIr; s/;Pn.x; r/ANDTn.x; r/

The following theorem gives connection in congruences between the polynomials wnandPn:

Theorem 3. Letn; rbe non-negative integers andpbe a prime number. Then, for m2 f0; : : : ; p 1gthere holds

p 1

X

kDm

. x/^kwn.xIrCk; k/

.k m/Š . 1/^mmŠ.rCm/ⁿPp 1.x; m/:

In particular, formD0;we get

p 1

X

kD0

. x/^kwn.xIrCk; k/

kŠ rⁿ.1CxC Cx^{p 1}/:

Proof. Fork < mwe gethmC1ip 1 kD0and formkp 1we have

hmC1ip 1 kD.mCp k 1/Š

mŠ D.p 1 .k m//Š

mŠ 1

mŠ

. 1/^{k m} .k m/Š: wherehxinDx.xC1/ .xCn 1/ifn1andhxi0D1:Then

LHS . 1/^mmŠ

p 1

X

kD0

hmC1ip 1 kx^kwn.xIrCk; k/

. 1/^mmŠ

p 1

X

kD0

hm pC1ip 1 k.w_xCr/ⁿ.w_x/_k

. 1/^mmŠ

p 1

X

kD0

p 1 k

!

hm pC1ip 1 k.w_xCr/ⁿh w_xik

D . 1/^mmŠhm pC1 wxip 1.wxCr/ⁿ D . 1/^mmŠ.w_x mCp 1/p 1.w_xCr/ⁿ

D . 1/^mmŠ.w_x mCrCm/ⁿ.w_x mCp 1/p 1

D . 1/^mmŠ

n

X

jD0

(nCrCm jCrCm )

rCm

.w_x m/j.w_x mCp 1/p 1:

But forj 1we have

.wx m/j.wx mCp 1/p 1D.wx mCp 1/jCp 1

.w_x m 1/jCp 1D.jCp 1/ŠPjCp 1.x; mC1/

ı_.jD0/Pp 1.x; mC1/;

hence, it followsLHS . 1/^mmŠ.rCm/ⁿPp 1.x; m/.

A connection in congruences between the polynomialswnandTnis to be:

Theorem 4. For any integersn; m; r0and any primep;there holds

p 1

X

kD0

. m/_{p 1 k}.xC1/^kwn.xIrCm; k/ rⁿTp 1.xIm/:

Proof. Upon using the identity.xC1/^swn.xIr; s/D.w_xCr/ⁿ.w_xCs/s and the known congruence.m/_{p 1 k} ^{p 1}_k

h mip 1 k we obtain

LHS

p 1

X

kD0

p 1 k

!

hmip 1 k.w_xCrCm/ⁿ.w_xCk/k

p 1

X

kD0

p 1 k

!

hmip 1 k.w_xCrCm/ⁿhw_xC1ik

D.w_xCrCm/ⁿhw_xCmC1i^{p 1} .w_xCmCr/ⁿ.w_xCmCp 1/p 1

D

n

X

jD0

(nCr jCr

)

r

.w_xCm/j.w_xCmCp 1/p 1

D

n

X

jD0

(nCr jCr

)

r

.w_xCmCp 1/jCp 1

D

n

X

jD0

(nCr jCr

)

r

.jCp 1/ŠTjCp 1.xIm j /

D.p 1/ŠTp 1.xIm/C

n

X

jD1

(nCr jCr

)

r

.jCp 1/ŠTjCp 1.xIm j /

rⁿTp 1.xIm/:

Corollary 8. LetRn;t.xIr; s/ be as in Corollary7. Then, for any non-negative integersn; m; r; sand any primep−m;there holds

p 1

X

kDm

. x/^k k m

!Rn;t.xIrCk; k/

kŠ . 1/^m.rCm/ⁿLt.x; rCm/Pp 1.x; m/:

Proof. Theorem3implies

LHS D

t

X

jD0

aj.x/

p 1

X

kDm

. x/^k k m

!wnCj.xIrCk; k/

kŠ

t

X

jD0

aj.x/. 1/^m.rCm/^nCjPp 1.x; m/

. 1/^m.rCm/ⁿLt.x; rCm/Pp 1.x; m/:

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Authors’ addresses

Miloud Mihoubi

USTHB, Faculty of Mathematics, RECITS Laboratory, P. O. Box 32 El Alia 16111 Algiers, Algeria E-mail address:mmihoubi@usthb.dz, miloudmihoubi@gmail.com

Said Taharbouchet

USTHB, Faculty of Mathematics, RECITS Laboratory, P. O. Box 32 El Alia 16111 Algiers, Algeria E-mail address:staharbouchet@usthb.dz, said.taharbouchet@gmail.com