Some congruences related to harmonic numbers and the terms of the second order sequences

The second order sequence {Wn (c, d; r, s)}, or briefly {Wn}, is defined for n > 0 by Wn+1 = rWn + sWn−1 in which W0 = c,W1 = d, where c, d, r, s are arbitrary integers. As some special cases of {Wn}, denote Wn (0, 1; r, 1), Wn (2, r; r, 1) by Un and Vn, respectively. When r = 1, Un = Fn (the nth Fibonacci number) and Vn = Ln (the nth Lucas number). If α and β are the roots of the equation x2−rx−1 = 0, the Binet formulas of the sequences {Un} and {Vn} have the forms


Introduction
The second order sequence {W n (c, d; r, s)}, or briefly {W n }, is defined for n > 0 by W n+1 = rW n + sW n−1 in which W 0 = c, W 1 = d, where c, d, r, s are arbitrary integers. As some special cases of {W n }, denote W n (0, 1; r, 1), W n (2, r; r, 1) by U n and V n , respectively. When r = 1, U n = F n (the nth Fibonacci number) and V n = L n (the nth Lucas number).
If α and β are the roots of the equation x 2 −rx−1 = 0, the Binet formulas of the sequences {U n } and {V n } have the forms U n = α n − β n α − β and V n = α n + β n , respectively. From [2,3], E. Kılıç and P. Stanica derived the following recurrence relations for the sequences {U kn } and {V kn } for k ≥ 0, n > 0. It is clearly that where the initial conditions of the sequences {U kn } and {V kn } are 0, U k , and 2, V k , respectively. Binet formulas of the sequences {U kn } and {V kn } are For a prime p and an integer a with a p, we write the Fermat quotient q p (a) = a p−1 − 1 /p. Let Z be the set of integers. Z p denote the set of those rational numbers whose denominator is not divisible by p and is called as the set of p-adic integer numbers. For an integer D, . It is clearly that x 2 − x − 1 has two simple roots in Z p if and only if p ≡ ±1 (mod p).
In [1], A. Granville showed the congruence x i i . In [4], H. Pan and Z. W. Sun showed the following lemma and proposition: Lemma 1.1. Let p > 3 be a prime. Then where y n = W n (2, r; r, −s) and γ, δ are the two roots of the equation In this paper, we investigate the congruences involving harmonic numbers and terms of second order sequences {U kn } and {V kn } . For example, for where ∆ = V 2 k + 4 (−1) k+1 , a prime number p > 3, and an integer k with p V k .

Some Lemmas
In this section, we need the following lemmas for further use.
Lemma 2.1. For n ∈ N and x ∈ R, we have the following sums: Proof. For the proof of (5), from the sum as claimed. Similarly, the other result is proven. Thus, this ends the proof.
Lemma 2.2. For n ∈ N and x ∈ R, we have the following sums: Proof. For the first claim, from the sums as claimed. The other claim is similarly obtained. Thus, the proof is completed.
Lemma 2.3. For n ∈ N and x ∈ R, we have the following sums: Proof. Considering the sums the proof is clearly given.
28 Some congruences where ∆ = V 2 k + 4 (−1) k+1 and Legendre symbol . p . Proof. For the proof of (11), using the Binet formula of the sequence {V kn } and taking α 2k where any p-adic integer x. We get Similarly, using Binet formula of the sequence {U kn }, the proof of the congruence in (10) is given.
Lemma 2.5. Let p > 3 be a prime. For an integer k with p V k and Proof. Consider (1), respectively, we write as claimed.
Lemma 2.6. Let p > 3 be a prime. For an integer k with p V k and Proof. Consider that For ∆ p = 1, by taking V k , (−1) k instead of r, s in (4), respectively, we have (12) and from Fermat's little theorem, the congruence 1 (p−k) 2 ≡ 1 k 2 (mod p) for k p and α k β k = (−1) k , we get By (12) and (13), we obtain the desired result.

The Results involving the terms of the sequences {U kn } and {V kn }
In this section, we give congruences for the terms of the sequences {U kn } and {V kn }. Now we start with our first result.
Now, we will give the congruences with harmonic numbers of order 2, H n,2 .
Theorem 3.4. Let p > 3 be a prime. For ∆ p = 1, Proof. From Binet formula of the sequence {V kn } , we consider By taking p instead of n and α k V k , β k V k instead of x in (6), respectively, we have From (22), (23) and the congruence H p−1,2 ≡ 0( mod p), we get Using Binet formula of the sequence {V kn } and Lemma 2.6, we have which settles the proof.
Proof. From Binet formula of the sequence {V kn } and α 2k β 2k = 1, we have

Some congruences
If we take p instead of n and α k V k , β k V k instead of x in (8), respectively, we get and From (24), (25) and the congruence H p−1,2 ≡ 0( mod p), we have By α k β k = (−1) k , we rewrite which, by Lemma2.5 and Lemma2.6, equivalents