MATHEMATICA MONTISNIGRI
Vol LVIII (2023)

ON CERTAIN CONGRUENCES INVOLVING CENTRAL BINOMIALCOEFFICIENTS
R. BOUMAHDI1, M. MIHOUBI2, AND L. KHALDI3*12

National Higher School of Mathematics, Algiers, Algeria
RECITS Laboratory, Faculty of Mathematics , USTHB, Algiers, Algeria
3
LIM Laboratory, Department of Mathematics , University of Bouira,10000 Bouira, Algeria*Corresponding author. E-mail: l.khaldi@univ-bouira.dz
DOI: 10.20948/mathmontis-2023-58-3
Summary. Let
be an odd prime. In this paper, using some properties of Fibonaccinumbers, reciprocal polynomials for Fibonacci polynomials, and Legendre symbol, weestablish some congruences involving central binomial coefficient modulo and . We also
give some new identities for hyperbolic functions.
1INTRODUCTION

The th central binomial coefficient
function as follows:,, can be defined by the generating

and the first ten central binomial coefficients are([1, A000984], in The On-Line
Encyclopedia of Integer Sequences (OEIS)). The study of the properties and congruences
related to the central binomial coefficients goes back to Morley [2] who in 1895 proved thatfor any odd prime , we havethis congruence has been extended later toby Carlitz [3, 4] by showing that

whereis the th Bernoulli number. After that, many generalizations have been obtained
during the last three decades in many directions [5, 6, 7].
If is an integer not divisible by an odd prime number, we say that is a quadratic
residue of if
for some integer , otherwise is a non-quadratic residue.

2020 Mathematics Subject Classification: 11А07, 11B39, 05A10.
Key words and Phrases: Congruence, Fibonaccci number, Fibonacci polynomial, Binomial coefficient.

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R. Boumahdi, M. Mihoubi and L. Khaldi

The Legendre symbol denoted byis defined as follows:

One of the interesting congruences of Legendre symbol in addition to the law of quadratic
reciprocity of Gauss is given by

The Fermat quotient of a positive integer
is defined by

The Fibonacci numberswith respect to an odd prime

not dividing

are defined by the recurrence relation

The Binet formulas for Fibonacci numbers are given by

whereandare the roots of the characteristic polynomial, from which it follows
that,and. For more information about Fibonacci sequences
and their applications, the reader is referred to [8, 6].
We have the following congruence relating the Legendre symbol and Fibonacci numbers

In 2011, Sun and Tauraso [7] proved that for a prime
holds true

the following congruence

In 2016, Ji-Cai Liu [9] proved the conjecture of Apagodu and Zeilberger [10] stating that

Recall that a -integer is a rational number whose denominator is co-prime withthe ring of -integers denoted
we write
whenBelow we present some congruences that we will prove.

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and inin .

R. Boumahdi, M. Mihoubi and L. Khaldi

Theorem 1.1. Let

be a prime. Then we have

Theorem 1.2. Let

be an odd prime. Then we haveandIn particular,

Theorem 1.3. Let

be a prime. Then we have

In particular,

Note that the congruence (6) is an extension of the congruence proved by Apagodu andZeilberger [10, Proposition 1] and the congruence (5) is an extension proved by Liu [9]. In thenext section, we give some properties of reciprocal polynomials and some lemmas, which will
be used in the proofs of the previous theorems as well as some identities of Section 4.
2 PRELIMINARIESFor any non-negative integer
defined by
The explicit formula of, let

be the Fibonacci polynomial sequenceis34

R. Boumahdi, M. Mihoubi and L. Khaldi

denoted by

The reciprocal polynomial of a polynomial
defined by

Then the reciprocal polynomial ofisLemma 2.1. ([11, Theorem 1]). For any non-negative integers

Lemma 2.2. Let

be an odd prime and, we have

. We haveThis congruence has been proven by Z.H. Sun [12].
Lemma 2.3. Let be an odd prime. Then we have

Proof. By using the binomial expansion, we have

35isR. Boumahdi, M. Mihoubi and L. Khaldi

Notice that for each

there holds

Combining (10) and (11), we obtain the congruence (9).
3

PROOFS OF THEOREMS 1.1, 1.2 AND 1.3

Proof of Theorem 1.1. By using Lemma 2.1, for, we obtain

applying congruence (8) of Lemma 2.2, we get36andand noting that

R. Boumahdi, M. Mihoubi and L. Khaldi

Proof of Theorem 1.2. Taking

andin identity (7), we obtain

From Lemmas 2.2 and 2.3, we get the congruence (2)Finally, setting

andin identity (7) and using the congruence (8) we get

By using the congruence (2) modulo
Proof of Theorem 1.3. By taking

Here, this implies thatand Fermat’s little theorem we obtain relation (4).
andin the relation (7), we obtain, using congruence (8), we get37

R. Boumahdi, M. Mihoubi and L. Khaldi

By using the congruence (5) modulo
4and Legendre symbol we obtain relation (6).

SOME IDENTITIES FOR HYPERBOLIC FUNCTIONSHyperbolic functions play the analogous role of the trigonometric functions, but definedusing hyperbola instead of circle with radius . The pointis replaced by thepointfrom the right half of the unit hyperbola. In this section, we give someidentities of these functions by using reciprocal polynomials of Fibonacci polynomials.
Theorem 4.1.
i) For any integer, we have, we have

In particular, forii) For any integer, we have

Proof of Theorem 4.1. First, we will prove the identity (12), we haveOn the other hand

We deduceFor, in relation (14) and the elementary relations, we obtain identity (12).

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and

R. Boumahdi, M. Mihoubi and L. Khaldi

Now, we deal with identity (13). In one hand we haveOn the other hand

We deduce

For, in relation (15) we obtain identity (13).

Corollary 4.1.1. For any integer

Proof. Putting
Corollary 4.1.2. For any integer, we havein relations (12) and (13), we get (16) and (17), respectively.
, we have39

R. Boumahdi, M. Mihoubi and L. Khaldi

Proof. For5in relations (12) and (13), we get (18) and (19), respectively.

CONCLUSIONS

Using elementary tools and some properties of Fibonacci numbers, reciprocal polynomials
for Fibonacci polynomials and Legendre symbol, we have established new congruences
concerning central binomial coefficients modulo p and modulo. We also have given somecomments for hyperbolic functions as applications of our results. By this approach, we thinkthat it would certainly be possible to generalize the congruences obtained in Theorem 1.2 andTheorem 1.3 to modulo
and modulo
to derive some remarkable identities and properties
concerning binomial coefficients and some classical sequences.
REFERENCES[1] N. J. A, Sloane et al., The On–Line Encyclopedia of Integers Sequences, publishedelectronically at (2022). https://oeis.org.
[2] F. Morley, “Note on the congruence, whereis a prime”, Ann.
of Math., 9, 168-170 (1895).
[3] L. Carlitz, “A theorem of Glaisher”, Canadian J. Math., 5, 306-316 (1953).
[4] L. Carlitz, “Note on a theorem of Glaisher”, J. London Math. Soc., 28, 245-246 (1953).
[5] S. Koparal and N. Ömür, “On congruences related to the central binomial coefficient,harmonic and Lucas numbers”, Turkish J. Math., 4, 973-985 (2016).
[6] A. M, Legendre, Essai sur la théorie des nombres, at (2010).
http://www.archive.org/details/essaisurlathorOIeg.
[7] Z.-W. Sun and R. Tauraso, “On some new congruences for binomial coefficients”, Int. J.
Number Theory, 7, 645-662 (2011).
[8] T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, New York (2001).
[9] C.C. Liu, “On two conjectural supercongruences of Apagodu and Zeilbeger”, J. Differ. Equ.
Appl., 22 (12), 1791-1799 (2016).
[10] M. Apagodu and D. Zeilberger, “Using the“Fershman’s Dream” to prove combinatorialcongruences”, Amer. Math. Monthly, 124 (7), 597-608 (2017).
[11] D. Jennings, “Some polynomial identities for the Fibonacci and Lucas numbers”, Fibonacci
Quart., 31 (2), 134-137 (1993).
[12] Z.-H. Sun, “Congruences concerning Legendre polynomials”, Proc. Amer. Math. Soc., 139(6), 1915-1929 (2011).

Received, January 26, 2023

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