MATHEMATICA MONTISNIGRI Vol XLVIII (2020)

SOME CONGRUENCES INVOLVING CATALAN, PELL AND FIBONACCI NUMBERS

S. KOPARAL1* AND N. ÖMÜR1

1 Kocaeli University Mathematics Department 41380 İzmit Kocaeli Turkey. *Corresponding author. E-mail: [email protected]

DOI: 10.20948/mathmontis-2020-48-2

Summary. In this paper, using some special numbers and combinatorial identities, we show some interesting congruences: for a prime p5 ,

(p-1)/2 C P 22p+2 k 7-p  /2 p ,  k  2 -4 +8- (mod p) k=0 8 k+1  p p p

(p+1)/2 C 5 k-1 ,  k  -qp  2  + (mod p) k=1 4 k k+1  6

(p+1)/2 C L 2p-2 5 5 91 k-1  3p - + - (mod p)  k p+2  , k=1 -16  k  k+1  k+2  2 p p 2 p 40

where Cn , Ln and Pn are the n th Catalan number, the th Lucas number and the th Pell . number, respectively.  denotes the Legendre symbol. p

1 INTRODUCTION

The Fibonacci sequence Fn  and the Lucas sequence Ln  are defined by the following recursions:

Fn+1 F n +F n - 1 and Ln+1 L n +L n - 1, n  0 , where F00  , F11  and L20  , L11  , respectively. If α and β are the roots of equation x2 - x -1 0 , the Binet formulas of the sequences and have the forms αnn - β F  and L  αnn +β , n α - β n respectively.

The Pell sequence Pn  and the Pell-Lucas sequence Qn  are defined recursively by

Pn+1 2P+P n n - 1 and Qn+1 =2Q n +Q n - 1 , , in which P00  , P11  and Q01 Q 2 , respectively. If γ and δ are the roots of equation

2010 Mathematics Subject Classification: 11B39, 05A10, 05A19. Key words and Phrases: Central binomial coefficients, congruences, Fibonacci numbers, Pell numbers.

10 S. Koparal and N. Ömür

2 x - 2x-1 0 , the Binet formulas of the sequences Pn  and Qn  have the forms γnn -δ P  and Q  γnn +δ , n γ -δ n respectively. The harmonic numbers have interesting applications in many fields of mathematics, such as number theory, combinatorics, analysis and computer science. Harmonic numbers H n are defined as for a positive integer n n 1 Hn   , k=1 k 3 11 25 where H0 . The first few harmonic numbers are 1, , , ,... 0 2 6 12 Some elementary combinatorial properties of the Catalan numbers are given in [2, 6, 8]. The Catalan numbers are given by 1 2n   2n   2n  Cn     -   , n N. n+1n   n   n+1  For a prime p and an integer a with pa | , we write the Fermat quotient

p-1 a qp  a    a - 1  /p. For an odd prime and an integer ,  denotes the Legendre p symbol defined by  0 if p a, a    1 ifa isa quadraticresiduemodp , p  1 ifa isa quadraticnonresiduemodp . In [3], E. Lehmer showed that for prime p  3,

Hp - 1  /2  - 2qp  2   mod p . (1.1) In [9], Z.W. Sun obtained that

p - 1  /2 1 2k m m - 4  ,  k  mod p  k=0 mpk  and p - 1  /2 12k m m - 4 m m - 4  ,  k  - mod p  k=0 m k+1  k 2 2 p where is an odd prime and m is any integer not divisible by p. Let be a fixed prime  3 . Define p xp -  x - 1  - 1 p - 1 xk qx and Gx ,   n    n p k=1 k where x is variable. In [1], A. Granville showed that

11 S. Koparal and N. Ömür

q x  - G1  x   mod p , p G2  x  G2  1- x  +x G2  1 - 1/x   mod p  ,

2 pp q x    -2x G22 x  -2 1 - x  G 1 - x   mod p  . It was proved in [1] (the last expression on page 3) that for any integer n1 , k k n n n - 1 n - 1 - x  1 n - 1  1 - x  -1 1 - x  -  - x  -1  + . 221.2  k=1 k - 1 k nk=1 k n In [7], Z.H. Sun obtained the following congruences: for an odd prime p and

Gnp x  Z x , pk 1 1+ x - 1  - xp p - 1  1 - x  -1 p - 1 xr r - 1 1 G x   - +p mod p2 , 2  2    p p k=1 kr=2 r s=1 s and for a prime p3 and nN , n p2 npGn+1  x   -1  x Gn  1/x  - Gn  x   mod p  . In [10], Z.W. Sun showed that for a prime , p - 1 HL p - 1 HFF2 p - 1  kk 0 mod p  and k k  k mod p . k=1 k k=1 k pk=1 k In [5], H. Pan and Z.W. Sun obtained that for a prime p5 , p - 1 L k  0 mod p ,  2   k=1 k and for a prime p 2,5 , pa - 1 2k a  k p 3 -1   1-2Fa  mod p  ,   a p k=0 k 5 p -  5 where a is a positive integer. In [4], S. Mattarei and R. Tauraso showed that for a prime , p - 12k xk  2G λ +G μ mod p , 2  22      k=1 k k 1 1 where λ= 1+ 1 - 4x and μ= 1- 1 - 4x . 2   2  

2 SOME CONGRUENCES INVOLVING SPECIAL NUMBERS In this section, we will give the congruences involving some special numbers. Now, we give the following lemma for further use.

Lemma 1. Let n be any positive integer. Then k+1 k+1 n+1 n n - 1 -x  1xn - 1 1 - x  -1 1 - x  -1 22= + + Hn k=1 k - 1 k k+1  nk=1 k k+1  n n+1  n

12 S. Koparal and N. Ömür

and k+2 k+2 n+2 n n - 1 -x  1 n - 1 1 - x  -1 1 - x  -1 x x2 22= + + - Hn . k=1 k - 1 k k+1  k+2  nk=1 k k+1  k+2  n n+1  n+2  n+1 2n

Proof. From Binomial Theorem, we have

k+1 n+1 n - 1 1 - x  -1 1 - x  -1  + +xHn k=1 k k+1  n n+1  k+1 k+1 n 1 - x  -1 n1 n 1 - x  -1+  k+1  x =+x  =  k=1 k k+1  k=1 k k=1 k k+1 

kj n+1 n+1 k n+1 k 1 - x  -1+kx 11k  j -x  k-1  ==   -x  =     k=2 k k -1  k=2 k k -1  j=2 j  k=2k -1 j=2 j j-1  j j n+1 k -x   k-2 n+1 -x  n1 k-2  = =   . k=2 j=2 j j-1   j-2 j=2 j j-1  kj j-2 

n+1 k-2   n  From the equality  =   , we get k=j j-2   j-1 

k+1 n+1 n - 1 1 - x  -1 1 - x  -1  + +xHn k=1 k k+1  n n+1 

j j+1 j+1 n+1 n -x  n n  -x  n n-1  -x  = = =n .        2 j=2 j-1 j j-1  j=1 j  j j+1  j=1 j-1  j j+1  Thus, the first identity is obtained. Similarly, we obtain other identity.

a For a prime p and a rational number x= written in reduced form, x 0  mod p  if and b only if a is divisible by .

Theorem 1. Let be an odd prime. For any rational number x such that x- 1  0 mod p  , p - 1  /2 p-1 C p+1 2  k xk+1  1-λ - μ  + λp +μ p - 1   mod p , (2.1) k=0 k+1 p 1 1 where λ= 1+ 1 - 4x and μ= 1- 1 - 4x . 2   2  

Proof. Replacing n and by (p+1)/2 and 4x in (1.2), respectively, we have

13 S. Koparal and N. Ömür

k+1 k p+1  /2 p+1 p - 1  /2 p - 1  /2 - 4x  p - 1  /2  1 - 4x   1 - 4x  -1

2 = +2 - Hp - 1  /2 . 2 k=0 k k+1  k=1 k p+1

p - 1  /2 k 2k From the congruences -4    mod p  for k=1,…, p - 1  /2 , k k 11  mod p  for k=1,…,p - 1 and (1.1), we write p+k k

p - 1  /2 k+1 p - 1  /2 k 2k x p+1  /2 1 - 4x  -2  2 1 - 4x - 1 + +2q 2 mod p . (2.2)  2    p     k=0 k k+1  k=1 k

p - 1 k By congruence  -1   mod p  for , we get k kk p - 1  /2 1 - 4x  p - 1  /2 p - 1 1 - 4x   -  (2.3) k=1 kkk=1 2k - 1 pp p - 1  /2 2 - 1+ 1 - 4x - 1 - 1 - 4x 2 p k      -  1 - 4x  = mod p . ppk=1 2k

Substituting (2.3) into (2.2), we have p - 1  /2 p - 1  /2 k+1 Ck k+1 2k x x=  2 k=0 k+1 k=0 k k+1  pp 1 - 4x - 1 - 1 - 4x+1 +2 p+1  /2      1 -  1 - 4x  - q 2  - p 2p p - 1 p+1 2 =1 - λ - μ  + λpp +μ - 1   mod p  . p Thus, this concludes the proof.

From Theorem 1, we immediately deduce the following results. Corollary 1. Let p be an odd prime. Then p - 1  /2 C k  k  4 1 - qp  2    mod p , k=0 4 k+1 

p - 1  /2 k -1  Ck p - 1 1 - Lp 5   2 +5 - 1 mod p  , (2.4) k=0 k+1 p p p - 1  /2 p+2 Ck 7 - p  /2P p 22  k  2 - 4 +8 -  mod p , k=0 8 k+1  p p p

14 S. Koparal and N. Ömür

p - 1  /2 p+1 Ck 2Qp 22  k  - + +8 - 4 mod p , k=0 -4   k+1  p p p

p - 1  /2 3 - p p+3 Ck 2 2 5  k  - L3p + +20 - 16 mod p  , k=0 -16   k+1  p p p and for p3 p - 1  /2 C 25p-1 L  k 2p  k  p-2 - 9 - 5 +9 mod p . k=0 9 k+1  p 3 p Proof. For the proof (2.4), considering x= -1 in Theorem 1, we have p - 1  /2 Ck  k+1 k=0 -1   k+1  pp p - 1      p-1 p+1  /2 2 1+ 5 1 - 5 5 2 1- 5 +  +   - 1 1- 5 + Lp - 1   mod p  . p 2   2  p p      Similarly, we obtain other five congruences of Corollary 1.

Theorem 2. Let be a prime. For any rational number x such that x -1  0 mod p  , p+1  /2 p - 1 C 2 x 8x+1p+1 6x+1  k - 1 xk+1  λp +μ p - 1  -  λ - μ  + mod p , k=1 k k+1  p 12 12 where  and  are defined as before. Proof. By the first identity of Lemma 1, we have k+2 k+1 n+1 n - 1 n - 1 -x  1xn - 1 1 - x  -1 1 - x  -1 = + + H .  2 2 n k=0 k k+1   k+2  nk=1 k k+1  n n+1  n Replacing n and by (p+1)/2 and 4x in the aboving sum, respectively, we have k+2 k+1 p+3  /2 p - 1  /2 p - 1  /2 -4x  2 p - 1  /2 1 - 4x  - 1 1 - 4x  - 1 8x 22= +8 + Hp+1  /2 . k=0 k k+1   k+2  p+1k=1 k k+1  p+1   p+3  p+1

p - 1  /2 k 2k From the congruences -4    mod p  for k=1,…, p - 1  /2 , k k 11  mod p  for k=1,…,p - 1 and (1.1), we write p+k k

15 S. Koparal and N. Ömür

p - 1  /2 2k xk+2   2 k=0 k k+1   k+2 

k p+1  /2 p+3  /2 1- 4x  - 1p - 1  /2  1 - 4x  1 1 - 4x   1 - 4x  x 1   + - + +x - - xqp  2  -  mod p . 8 k=1 k 4 p+1  4 p+1  6 2 6 1 From the congruences (2.3) and  1 mod p  , we have p+1 p+1  /2 C  k - 1 xk+1 k=1 k k+1  p+1  /2 p+3  /2 x pp1 - 4x   1 - 4x  x 5  - 2 - 1+ 1 - 4x - 1 - 1 - 4x - + + - xq 2  - 2p       4 6 2 p 2 p+1  /2 p+3  /2 x pp1 - 4x   1 - 4x  x 5 =- 2p - 1+ 1 - 4x - 1 - 1 - 4x - + + -  mod p . 2p       4 6 2 2 Thus, the desired result is obtained. From the above equation, we get the assertion of the theorem.

As an immediate consequence of Theorem 2, we obtain the following result. Corollary 2. Let p3 be a prime. Then

p+1  /2 C 5 k - 1  k  - qp  2  + mod p , k=1 4 k k+1  6

p+1  /2 k -1  Ck - 1 p - 1 Lp - 1 35 5 5   2 -  + mod p , k=1 k k+1  p 12 p 12

p+1  /2 p - 1 p+1  /2 C Pp 2 4 1 7 k - 1  - - + mod p ,  k p - 1  /2    k=1 8 k k+1  2pp 3 2 6 p+1  /2 C 2 p - 1 L 85 5 5 k - 1 2p  kp  - 1 -  + mod p , k=1 9 k k+1  p 3 108 p 4 p+1  /2 p Ck - 1 Qp - 2 2 2 1  k  -  + mod p , k=1 -4  k  k+1  2p 3 p 6 and p+1  /2 p - 1 Ck - 1 L3p 2 5 5 5  k  p+1 - + -  mod p . k=1 -16  k  k+1  2 p p 6 p 6 Theorem 3. Let p5 be a prime. For any rational number x such that x -1  0 mod p  ,

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p+1  /2 p - 2 C 21p+5 k - 1 xxk+2  2λ p +μ p -1  -  λ - μ   k k+1 k+2 p 60 k=1    5 p+3 1 p+1 1 1 1 + λ - μ  - λ - μ  + x2 + x -  mod p , 96 32 8 12 240 where  and  are defined as before.

As an direct consequence of Theorem 3, we obtain the following result.

Corollary 3. Let p5 be a prime. Then

p+1  /2 C 1 47 k - 1  k  - qp  2  + mod p , k=1 4 k k+1  k+2  2 120

p+1  /2 k -1  Ck - 1 p - 2 Lp - 1 15 5 3   2 -  + mod p , k=1 k k+1  k+2  p 16 p 80

p+1  /2 p - 2 p+1  /2 C Pp 2 3 1 21 k - 1  - - + mod p ,  k p+1  /2    k=1 8 k k+1  k+2  2pp 5 2 40

p+1  /2 p - 2 p+1  /2 C 2 L2p 29 5 43 k - 1  - 1 - + mod p ,  kp   k=1 9 k k+1  k+2  p 3 48 9 80 p+1  /2 p Ck - 1 Qp - 2 1 2 11  k  + -  mod p , k=1 -4  k  k+1  k+2  4p 5 p 40 and p+1  /2 p-2 Ck - 1 L3p 2 5 5 91  k  p+2 - + -  mod p . k=1 -16  k  k+1  k+2  2 p p 2 p 40

3. CONCLUSION

Using combinatorial identities, some interesting congruences are investigated on the sums which include Catalan numbers. It is conceivable to extend our results using the useful technics or methods. A first question on the extension of these congruences can be viewed as generalizations on using the -Binomial coefficients instead of the binomial coefficients. A second question on such extensions of these congruences can be observed that for an odd prime p and any rational number x such that x -1  0 mod p  , the congruences of the sums p - 1  /2 C k - 1 xk (mod p). with decreasing factorials including Catalan numbers  m k=1 k

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Received April 11, 2020

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