For example, for an odd primep¤5, .p 1/=2 P kD0

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 2, pp. 1079–1094 DOI: 10.18514/MMN.2018.1222

SOME INTERESTING CONGRUENCES FOR BALLOT NUMBERS

NES¸E ¨OM ¨UR AND SIBEL KOPARAL Received 30 April, 2014

Abstract. In this paper, we determine the sums

n 1

P

kD0 2kCd

k

=x^k;

n 1

P

kD0

k ^2k_k^Cd

=x^k and some congruences can be obtained by using them. For example, for an odd primep¤5,

.p 1/=2

P

kD0

. 1/^kk ^2k_k^{Cd . 1/}

d 5

5 p

FdC1 ₅ p

.dC1/ L

dC1 ₅ p

.modp/;

and for an odd primep;

.p 1/=2

P

kD0 k

. 4/^kB .k; d / . 2/^{d 1}

2 p

dPdC1 ₂ p

.modp/;

whered2 f0; :::; .p 1/ =2g; Fnis thenth Fibonacci number,Lnis thenth Lucas number and Pnis thenth Pell number.

: p

denotes the Legendre symbol.

2010Mathematics Subject Classification: 11B65; 05A10; 11A07 Keywords: congruences, binomial coefficients, Ballot numbers

1. INTRODUCTION

The some elementary combinatorial properties of the Catalan and Ballot numbers are given in [2], [4] and [3]. In [1], [7], ´E. Lucas and N.J. Fine gave how to compute binomial coefficients modulo a prime. Lucas Theorem is given as follows:

Ifp is a prime,n; m; n0 andm0are non-negative integers, andn0,m0 are both less thanp;then

npCn0

mpCm0

! n

m

! n0

m0

!

.mod p/: (1.1)

The Catalan numbers are given by CnD 1

nC1 2n

n

!

D 2n n

! 2n nC1

!

; n2N:

c 2018 Miskolc University Press

In [2], the Catalan numbers are special cases of the Ballot numbers B .n; k/D k

2nCk

2nCk n

! :

In [13], Z.W. Sun and R. Tauraso obtained

p^a 1

P

kD0 2k kCd

=m^k and

p 1

P

kD0 2k kCd

=km^{k 1} modpfor alldD0; 1; :::; p^a;wheremis any integer not divisible byp. For example, they showed that ifp¤2; 5;then

p 1

X

kD0

. 1/^k 2k k

! 5

F_p .^p₅/

p .mod p/:

In [8], Z.W. Sun determined

p^a 1

P

kD0 2k kCd

=m^k mod p²fordD0; 1Ifor example,

p^a 1

X

kD0

2k kCd

!

=m^k

m² 4m p^a

C

m² 4m p^{a 1}

up

m² 4m p

.modp²/;

where an odd primepanda; m2Zwitha > 0,p−m.

In [11], Z.W. Sun used Lucas quotients in order to obtain

.p 1/=2

P

kD0 2k

k

=m^kmodulo p²for any integerm60 .mod p/; especially, he determined the following congru- ence:

.p 1/=2

X

kD0 2k

k

16^k

3 p

.modp²/:

In [9], Z.W. Sun gave the following congruence:

.p 3/=2

X

kD0

2k k

.2kC1/ 4^k . 1/^.pC1/=2qp.2/ .mod p²/;

where an odd primepandqp.2/is a Fermat quotient.

In [6], S. Koparal and N. ¨Om¨ur presented congruences involving central binomial coefficients and harmonic numbers. For example, for an odd primep,

.p 1/=2

X

kD0

. 1/^k 2k k

!

H_{k 1}2^p p

2FpC1 5^{.p 1/=2} 1

.modp/;

whereHnis thenth harmonic number.

In [5], K.H. Pilehrood et all gave that for a primep¤2; 5;

.p 3/=2

X

kD0 2k

k

F_2kC1

.2kC1/ 16^k . 1/^.pC1/=2Fp p 5

p .modp²/:

In this paper, we determine the sums

n 1

P

kD0 2kCd

k

=x^k;

n 1

P

kD0

k ^2k_k^Cd

=x^k and some congruences can be obtained by using them.

Two sequencesfun.x/gandfvn.x/gof polynomials are defined by forn > 0 unC1.x/Dxun.x/ un 1.x/ and vnC1.x/Dxvn.x/ vn 1.x/ ; whereu0.x/D0; u1.x/D1 andv0.x/D2; v1.x/Dx; respectively. The char- acteristic equationy² xyC1D0of the sequencesfu_n.x/gandfv_n.x/ghas two roots

˛ .x/D xCp

2 andˇ .x/Dx p 2 ;

whereDx² 4. The Binet formulas of the sequencesfun.x/gandfvn.x/gare as follows:

un.x/D.˛ .x//ⁿ .ˇ .x//ⁿ

˛ .x/ ˇ .x/ andvn.x/D.˛ .x//ⁿC.ˇ .x//ⁿ; respectively. Clearly, for anyn2N;

xun.x/Cvn.x/D2unC1.x/ ; (1.2) un.x/CxC2

vn.x/D 2

.vnC1.x/Cvn.x// ; (1.3) un.x/DvnC1.x/ vn 1.x/ : (1.4) It is seen that

. 1/^{n 1}u_n. 3/DF_2nand. 1/ⁿv_n. 3/DL_2n; (1.5) . 1/^{n 1}un. 6/D1

2P2nand. 1/ⁿvn. 6/DQ2n;

whereFnandLnare thenth Fibonacci number andnth Lucas number, andPnand Qnare thenth Pell number and thenth Pell-Lucas number, respectively.

2. SOME CONGRUENCES RELATED TOBALLOT NUMBERS

In this section, we will investigate some congruences with the combinatorial iden- tities. Now, we give the following lemmas for further use.

Lemma 1. Letr; s2ZandDDr² 4s:Suppose that p is an odd prime with p−sD. Then

r˙p D 2

!p

D p

s

1

D p

=2 .modp/; (2.1)

where ^r˙

pD

2 are roots of the equationy² ryCsD0[10]:

Lemma 2. For anyn2N, we have d

dx.un.x 2//D 1

..nC1/ vn.x 2/ 2unC1.x 2// ; whereas before.

Proof. By differentiating both sides of˛ .x 2/D^{x 2C}

p

2 ;we write d

dx.˛ .x 2//Dx 2Cp 2p

D˛ .x 2/

p : Similarly, it is clearly seen that

d

dx.ˇ .x 2//D ˇ .x 2/

p and d dx

p

D2˛ .x 2/

p 1D2ˇ .x 2/

p C1:

Thus d

dx.un.x 2//D D d

dx

˛ⁿ.x 2/ ˇⁿ.x 2/

p

D 1

n˛ⁿ.x 2/

p Cnˇⁿ.x 2/

p p

2˛ .x 2/

p 1

˛ⁿ.x 2/C

2ˇ .x 2/

p C1

ˇⁿ.x 2/

D 1

n˛ⁿ.x 2/Cnˇⁿ.x 2/ 2

˛^nC1.x 2/ ˇ^nC1.x 2/

p

C˛ⁿ.x 2/Cˇⁿ.x 2/

D 1

.nvn.x 2/ 2unC1.x 2/Cvn.x 2//

D 1

..nC1/ vn.x 2/ 2unC1.x 2// :

This concludes the proof.

Theorem 1. For anyn,d 2Z^C, we have

n 1

X

kD0

2kCd k

!

x^{n 1 k}D

nCd 1

X

kD0

2nCd k

!

u_{nCd k}.x 2/ (2.2)

x^nCbd=2c "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

; where"D

1 . 1/^d

=2:

Proof. To prove (2.2), we shall apply induction method onn.

FornD1;we must show that ford2Z^C;

d

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D (2.3)

D1Cx^bd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

: We have

d

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D

D1C

dC2

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D1C

dC2

X

kD0

dC2 k

!

u_{k 1}.x 2/

D1C 1

˛ .x 2/ ˇ .x 2/

8

<

: 1

˛ .x 2/

dC2

X

kD0

dC2 k

!

.˛ .x 2//^k 1

ˇ .x 2/

dC2

X

kD0

dC2 k

!

.ˇ .x 2//^k 9

=

; D1C 1

˛ .x 2/ ˇ .x 2/

1

˛ .x 2/.1C˛ .x 2//^dC2 1

ˇ .x 2/.1Cˇ .x 2//^dC2

: Using the identity˛ .x 2/ ˇ .x 2/D1, we get

d

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D D1C 1

˛ .x 2/ ˇ .x 2/

n

.ˇ .x 2// .1C˛ .x 2//^dC2 .˛ .x 2// .1Cˇ .x 2//^dC2o

:

FromdD bd=2c C bd=2c C";we rewrite

d

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D D1C 1

˛ .x 2/ ˇ .x 2/

n

.ˇ .x 2// .1C˛ .x 2//^".1C˛ .x 2//^bd=2cC1.1C˛ .x 2//^bd=2cC1 .˛ .x 2// .1Cˇ .x 2//^".1Cˇ .x 2//^bd=2cC1.1Cˇ .x 2//^bd=2cC1o

: Using the identity.1C˛ .x 2// .1Cˇ .x 2//Dx;we get

d

X

kD0

dC2 k

!

u_{dC1 k}.x 2/D D1C 1

˛ .x 2/ ˇ .x 2/

n

.ˇ .x 2/C"/ .1C˛ .x 2//^bd=2cC1.1C˛ .x 2//^bd=2cC1 .˛ .x 2/C"/ .1Cˇ .x 2//^bd=2cC1.1Cˇ .x 2//^bd=2cC1o D1C x^bd=2cC1

˛ .x 2/ ˇ .x 2/

n

.˛ .x 2//^bd=2c .ˇ .x 2//^bd=2c C" .˛ .x 2//^bd=2cC1 " .ˇ .x 2//^bd=2cC1

o

D1Cx^bd=2cC1 u_bd=2c.x 2/C"u_bd=2cC1.x 2/

: So, (2.2) holds fornD1.

We assume that the result is true for some integern1:

We must show that fornC1; (2.2) holds. By the induction hypothesis, for any d 2Z^C;we write

n

X

kD0

2kCd k

!

x^{n k}D

D 2nCd n

! Cx

n 1

X

kD0

2kCd k

!

x^{n 1 k}

D 2nCd n

! Cx

nCd 1

X

kD0

2nCd k

!

u_{nCd k}.x 2/

x^nC1Cbd=2c "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

D 2nCd n

! Cx

nCd 1

X

kD0

2nCd k

!

u_{nCd k}.x 2/

2

nCd 1

X

kD0

2nCd k

!

u_{nCd k}.x 2/C2

nCd 1

X

kD0

2nCd k

!

u_{nCd k}.x 2/

nCd 1

X

kD0

2nCd k

!

u_{nCd 1 k}.x 2/C

nCd 1

X

kD0

2nCd k

!

u_{nCd 1 k}.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

D.x 2/

nCd

X

kD0

2nCd k

!

u_{nCd k}.x 2/

nCd

X

kD0

2nCd k

!

u_{nCd 1 k}.x 2/

C2

nCd 1

X

kD0

2nCd k

!

unCd k.x 2/C

nCd 1

X

kD0

2nCd k

!

unCd 1 k.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

:

Since.x 2/ un.x 2/DunC1.x 2/Cun 1.x 2/, we have

n

X

kD0

2kCd k

!

x^{n k} D

D

nCd

X

kD0

2nCd k

!

unCdC1 k.x 2/C2

nCd 1

X

kD 1

2nCd k

!

unCd k.x 2/

C

nCd 1

X

kD 2

2nCd k

!

u_{nCd 1 k}.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

D

nCd

X

kD0

2nCd k

!

u_{nCdC1 k}.x 2/C2

nCd

X

kD0

2nCd k 1

!

u_{nCdC1 k}.x 2/

C

nCdC1

X

kD0

2nCd k 2

!

u_{nCdC1 k}.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

;

and hence

n

X

kD0

2kCd k

!

x^{n k}D

D

nCd

X

kD0

2nCd k

!

C2 2nCd k 1

!

C 2nCd k 2

!!

u_{nCdC1 k}.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

: By the binomial identity ²ⁿ_k^Cd

C2 ²ⁿ_{k 1}^Cd

C ²ⁿ_{k 2}^Cd

D ^2nC_k^dC2

;we get

n

X

kD0

2kCd k

!

x^{n k}D

nCd

X

kD0

2 .nC1/Cd k

!

u_{nCdC1 k}.x 2/

x^nCbd=2cC1 "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

:

Hence the result is true for all integersn0:

As a result of Theorem1, we may give the following congruence.

Corollary 1. Letpbe an odd prime. Then

.p 1/=2

X

kD0

1 m^k

2kCd k

!

m^{bd=2c .p 1/=2} (2.4)

˚

.1 "/ u_{.p 1/=2Cbd=2c}.m 2/C..m 1/"C1/u_{.pC1/=2Cbd=2c}.m 2/

m^bd=2cC1."u_bd=2cC1.m 2/Cu_bd=2c.m 2// .mod p/;

where"; as before,d 2 f0; 1; :::; .p 1/ =2gandm2Zwithp−m:

Proof. SubstitutingnD.pC1/=2andxDmin (2.2), we write

.p 1/=2

X

kD0

2kCd k

!

m^{.p 1/=2 k}D

.p 1/=2Cd

X

kD0

pCdC1 k

!

u.pC1/=2Cd k.m 2/

m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

: By the congruence in (1.1), it is easily seen that

m^{.p 1/=2}

.p 1/=2

X

kD0

2kCd k

m^k

dC1

X

kD0

dC1 k

!

u_{.pC1/=2Cd k}.m 2/

m^{bd=2cC.pC1/=2} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

D

dC1

X

kD0

dC1 k

!

u_{.p 1/=2Ck}.m 2/

m^{bd=2cC.pC1/=2} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

D .˛ .m 2//^{.p 1/=2}

˛ .m 2/ ˇ .m 2/

dC1

X

kD0

dC1 k

!

.˛ .m 2//^k .ˇ .m 2//^{.p 1/=2}

˛ .m 2/ ˇ .m 2/

dC1

X

kD0

dC1 k

!

.ˇ .m 2//^k m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

.modp/:

By Binomial theorem, we have m^{.p 1/=2}

.p 1/=2

X

kD0

2kCd k

m^k

.˛ .m 2//^{.p 1/=2}.1C˛ .m 2//^dC1

˛ .m 2/ ˇ .m 2/

.ˇ .m 2//^{.p 1/=2}.1Cˇ .m 2//^dC1

˛ .m 2/ ˇ .m 2/

m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

.mod p/:

FromdD bd=2c C bd=2c C", then

.p 1/=2

X

kD0

2kCd k

m^k

1

˛ .m 2/ ˇ .m 2/

n

.˛ .m 2//^{.p 1/=2}.1C˛ .m 2//^bd=2cC"C1.1C˛ .m 2//^bd=2c .ˇ .m 2//^{.p 1/=2}.1Cˇ .m 2//^bd=2cC"C1.1Cˇ .m 2//^bd=2co m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

.modp/:

Using the equalities of˛ .m 2/ ˇ .m 2/D1and.1C˛ .m 2// .1Cˇ .m 2//D

m, we have the proof.

For example, whend D0andm2Zwithp−m

.p 1/=2

X

kD0 2k

k

m^k

m .m 4/

p

.mod p/[12],

and

.p 1/=2

X

kD0

1 . 4/^k

2kCd k

!

. 2/^d 2

p

P_d 2 p

.modp/;

whered 2 f0; 1; :::; .p 1/ =2g.

Theorem 2. For anyn,d 2Z^C, we have

n 1

X

kD0

k 2kCd k

!

x^{n 1 k}D

nCd 1

X

kD0

2nCd k

!

..n 1/ unCd k.x 2/ (2.5) x

..nCdC1 k/ v_{nCd k}.x 2/ 2u_{nCdC1 k}.x 2//

C.bd=2c C1/ x^nCbd=2c "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

Cx^nCbd=2cC1

" .bd=2c C2/ v_bd=2cC1.x 2/ 2u_bd=2cC2.x 2/

C.bd=2c C1/ v_bd=2c.x 2/ 2u_bd=2cC1.x 2/

; where"; as before.

Proof. To prove equality (2.5), we take the derivative of (2.2) with respect to x:

n 1

X

kD0

2kCd k

!

.n 1 k/ x^{n 2 k}D

D

nCd 1

X

kD0

2nCd k

!1

..nCdC1 k/ vnCd k.x 2/ 2unCdC1 k.x 2//

.nC bd=2c/ x^{nCbd=2c 1} "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

x^nCbd=2c

" .bd=2c C2/ v_bd=2cC1.x 2/ 2u_bd=2cC2.x 2/

C.bd=2c C1/ v_bd=2c.x 2/ 2u_bd=2cC1.x 2/

; and then

n 1

X

kD0

k 2kCd k

!

x^{n 2 k}D (2.6)

Dn 1 x

n 1

X

kD0

2kCd k

!

x^{n 1 k} 1

nCd 1

X

kD0

2nCd k

!

..nCdC1 k/ vnCd k.x 2/ 2unCdC1 k.x 2//

C.nC bd=2c/ x^{nCbd=2c 1} "u_bd=2cC1.x 2/Cu_bd=2c.x 2/

Cx^nCbd=2c

" .bd=2c C2/ v_bd=2cC1.x 2/ 2u_bd=2cC2.x 2/

.bd=2c C1/ v_bd=2c.x 2/ 2u_bd=2cC1.x 2/

:

Multiplying both sides of (2.6) withxand using Lemma1, the proof is the complete.

Now, from Theorem2, we have the following congruence:

Corollary 2. Letpbe an odd prime. Then

.p 1/=2

X

kD0

k

2kCd k

m^k (2.7)

m^{bd=2cC1 .p 1/=2}

(2.8)

.dC2/ v_{.pC1/=2Cbd=2c}.m 2/C"v_{.pC1/=2Cbd=2cC1}.m 2/

2 .1 "/ u_{.pC1/=2Cbd=2c}.m 2/C..m 1/ "C1/ u_{.pC1/=2Cbd=2cC1}.m 2/

Cm^bd=2cC"C1

2 .bd=2c C1/ v_bd=2cC1.m 2/C.1 "/ v_bd=2c.m 2/

.2 "/ mu_bd=2cC1.m 2/

.modp/;

where"; as before,d 2 f0; 1; :::; .p 1/ =2gandm2Zwithp−m:

Proof. SubstitutingnD.pC1/=2andxDmin (2.5), we write

.p 1/=2

X

kD0

k 2kCd k

!

m^{.p 1/=2 k}D

D

.p 1/=2Cd

X

kD0

pCdC1 k

! p 1

2 u_{.pC1/=2Cd k}.m 2/

m

pC1

2 CdC1 k

v_{.pC1/=2Cd k}.m 2/ 2u_{.pC1/=2CdC1 k}.m 2/

C.bd=2c C1/ m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

Cm^{.pC1/=2Cbd=2cC1}

" .bd=2c C2/ v_bd=2cC1.m 2/ 2u_bd=2cC2.m 2/

C.bd=2c C1/ v_bd=2c.m 2/ 2u_bd=2cC1.m 2/

:

By the congruence in (1.1), it is easily seen that m^{.p 1/=2}

.p 1/=2

X

kD0

k ^2k_k^Cd

m^k

dC1

X

kD0

dC1 k

! 1

2u_{.pC1/=2Cd k}.m 2/

m

3

2Cd k

v_{.pC1/=2Cd k}.m 2/ 2u_{.pC1/=2CdC1 k}.m 2/

C.bd=2c C1/ m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

Cm^{.pC1/=2Cbd=2cC1}

" .bd=2c C2/ v_bd=2cC1.m 2/ 2u_bd=2cC2.m 2/

C.bd=2c C1/ v_bd=2c.m 2/ 2u_bd=2cC1.m 2/

D 1 2

dC1

X

kD0

dC1 k

!

u_{.p 1/=2Ck}.m 2/C2m

dC1

X

kD0

dC1 k

!

u_.pC1/=2Ck.m 2/

m

dC1

X

kD0

dC1 k

! kC1

2

v_{.p 1/=2Ck}.m 2/

C.bd=2c C1/ m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

Cm^{.pC1/=2Cbd=2cC1}

" .bd=2c C2/ v_bd=2cC1.m 2/ 2u_bd=2cC2.m 2/

C.bd=2c C1/ v_bd=2c.m 2/ 2u_bd=2cC1.m 2/

.mod p/:

From the Binet formulae of the sequencesfun.m 2/gandfvn.m 2/gand Bino- mial theorem, we have

m^{.p 1/=2}

.p 1/=2

X

kD0

k ^2k_k^Cd

m^k

1

2m^bd=2c .1 "/ u.p 1/=2Cbd=2c.m 2/C..m 1/ "C1/ u.pC1/=2Cbd=2c.m 2/

C 2

m^bd=2cC1 .1 "/ u.pC1/=2Cbd=2c.m 2/C..m 1/ "C1/ u.pC1/=2Cbd=2cC1.m 2/

m^bd=2cC1

.dC1/ v.pC1/=2Cbd=2c.m 2/C"v.pC1/=2Cbd=2cC1.m 2/

m^bd=2cC1

2 .1 "/ v.p 1/=2Cbd=2c.m 2/C..m 1/ "C1/ v.pC1/=2Cbd=2c.m 2/

C.bd=2c C1/ m^{.pC1/=2Cbd=2c} "u_bd=2cC1.m 2/Cu_bd=2c.m 2/

Cm^{.pC1/=2Cbd=2cC1}

" .bd=2c C2/ v_bd=2cC1.m 2/ 2u_bd=2cC2.m 2/

C.bd=2c C1/ v_bd=2c.m 2/ 2u_bd=2cC1.m 2/

.modp/:

From (1.2), (1.3) and (1.4), we obtained the desired result.

For example, for an odd primep¤5

.p 1/=2

X

kD0

. 1/^kk 2kCd k

!

. 1/^d 5

5

p F

dC1

5 p

.dC1/ L

dC1

5 p

.modp/;

and for an odd primep;

.p 1/=2

X

kD0

k . 4/^k

2kCd k

!

. 2/^{d 2} 2

p P

dC1 2

p

dC1

2 Q

dC1 2

p

.mod p/;

whered 2 f0; 1; :::; .p 1/ =2g:

From Corollary1and Corollary2, clearly the congruences are given as follows:

Corollary 3. Letp be an odd prime andd 2 f0; 1; 2; :::; .p 1/ =2g. For m2Z withp−m, then

.p 1/=2

X

kD0

B .k; d /

m^k

m^b.dC1/=2c .p 1/=2 "˚

u.p 1/=2Cb.dC1/=2c ".m 2/ u.pC1/=2Cb.dC1/=2c.m 2/

m^b.dC1/=2c.u_b.d 1/=2c.m 2/ u_b.d 1/=2cC2 ".m 2// ı0;d .modp/;

and

.p 1/=2

X

kD0

k

m^kB .k; d / m^b.dC1/=2c

d˚

m^{1 "} v_b.dC1/=2c ".m 2/ v_b.dC1/=2cC1.m 2/

m ^{.p 1/=2} v.pC1/=2Cb.d 1/=2c.m 2/ v.pC1/=2Cb.dC1/=2cC1 ".m 2/o

.modp/;

where"; as before andıi;j is the Kronecker delta.

Proof. Using the binomial identities ⁿ_k

D_k^{n n 1}_{k 1}

and ⁿ_k

D ^{n 1}_k

C ^{n 1}_{k 1} , we get

.p 1/=2

X

kD0

B .k; d / m^k D

.p 1/=2

X

kD0

1 m^k

d 2kCd

2kCd k

! D

.p 1/=2

X

kD0

1 m^k

d k

2kCd 1 k 1

! : Fork; d2Z^C, it is known that

2kCd 1 k

!

D 2kCd 1 k 1

! Cd

k

2kCd 1 k 1

! : So, we have

.p 1/=2

X

kD0

B .k; d / m^k D

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! 2kCd 1 k 1

!!

D

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! _{.p 1/=2} X

kD0

1 m^k

2kCd 1 k 1

!

D

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! _{.p 3/=2} X

kD 1

1 m^kC1

2kCdC1 k

!

D

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! 1 m

.p 3/=2

X

kD0

1 m^k

2kCdC1 k

! d 1 1

!

D

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! 1 m

.p 1/=2

X

kD0

1 m^k

2kCdC1 k

! ı_0;d

C 1 m^.pC1/=2

pCd .p 1/ =2

! : Since _{.p 1/=2}^pCd

0 .modp/ford2 f0; 1; 2; :::; .p 1/ =2g, from (1.1), we get

.p 1/=2

X

kD0

B .k; d /

m^k

.p 1/=2

X

kD0

1 m^k

2kCd 1 k

! 1 m

.p 1/=2

X

kD0

1 m^k

2kCdC1 k

!

ı_0;d .mod p/:

So, takingd 1 anddC1instead ofd in (2.4), respectively, this concludes the

proof.

For example, for an odd primep¤5

.p 1/=2

X

kD0

. 1/^kB .k; d /. 1/^dC1 5

p

Ld

5 p

ı_0;d .mod p/;

and for an odd primep;

.p 1/=2

X

kD0

k

. 4/^kB .k; d / . 2/^{d 1} 2

p

dPdC1

2 p

.mod p/;

whered 2 f0; 1; 2; :::; .p 1/ =2g:

REFERENCES

[1] N. Fine, “Binomial coefficients modulo a prime,”Amer. Math. Monthly, vol. 54, pp. 589–592, 1947.

[2] I. R. Gessel, “Super ballot numbers,”J. Symbolic Computation, vol. 14, pp. 179–194, 1992.

[3] P. Hilton and J. Pedersen, “The ballot problem and catalan numbers,”Nieuw Arch. Wisk., vol. 8, pp. 209–216, 1990.

[4] P. Hilton and J. Pedersen, “Catalan numbers, their generalization, and their uses,”The Mathemat- ical Intelligencer, vol. 13, pp. 64–75, 1991.

[5] T. P. K.H. Pilehrood and R. Tauraso, “Congruences concerning jacobi polynomials and ap˙ery-like formulae,”Int. J. of Number Theory, vol. 8, pp. 1789–1811, 2012.

[6] S. Koparal and N. ¨Om¨ur, “On congruences related to central binomial coefficients, harmonic and lucas numbers,”Turkish J. Math., vol. 40, pp. 973–985, 2016.

[7] E. Lucas, “Sur les congruences des nombres eul´eriens et des coefficients diff´erentiels des fonctions trigonom´etriques, suivant un module premier,”Bull. Soc. Math. France, vol. 6, pp. 49–54, 1877–

1878.

[8] Z. Sun, “Binomial coefficients, catalan numbers and lucas quotients,”Sci. China Math., vol. 53, pp. 2473–2488, 2010.

[9] Z. Sun, “On congruences related to central binomial coefficients,”J. Number Theory, vol. 131, pp.

2219–2238, 2011.

[10] Z. Sun, “On harmonic numbers and lucas sequences,”Publ. Math Debrecen, vol. 80, pp. 1–17, 2012.

[11] Z. Sun, “Fibonacci numbers modulo cubes of primes,”Taiwanese J. Math., vol. 17, pp. 1523–

1543, 2013.

[12] Z. Sun, “Congruences involving generalized central trinomial coefficients,” Sci. China Math., vol. 57, pp. 1375–1400, 2014.

[13] Z. Sun and R. Tauraso, “New congruences for central binomial coefficients,”Adv. in Appl. Math., vol. 4, pp. 125–148, 2010.

Authors’ addresses

Nes¸e ¨Om ¨ur

Kocaeli University, Mathematics Department, 41380 Kocaeli, Turkey E-mail address:neseomur@kocaeli.edu.tr

Sibel Koparal

Kocaeli University, Mathematics Department, 41380 Kocaeli, Turkey E-mail address:sibel.koparal@kocaeli.edu.tr