Evaluation of sums involving products of Gaussian q-binomial coefficients with applications

EVALUATION OF SUMS INVOLVING PRODUCTS OF GAUSSIAN q-BINOMIAL COEFFICIENTS WITH APPLICATIONS

EMRAH KILIC¸ * AND HELMUT PRODINGER**

Abstract. Sums of products of two Gaussian q-binomial coefficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences.

1. Introduction

Throughout this paper we use the following notations: the q-Pochhammer symbol (z; q)n =

(1 − z)(1 − zq) . . . (1 − zqn−1_{) and the Gaussian q-binomial coefficients}

n k  z,q = (z; q)n (z; q)k(z; q)n−k . When z = q, we use the notations (q)n and

n k  q instead of (q; q)n and n k  q,q, respectively. We

conveniently adopt the notation thatn_k_q = 0 if k < 0 or k > n. Define the {Un, Vn} sequences as linear recurrences for n ≥ 2 by

Un= pUn−1+ Un−2, U0= 0, U1= 1,

Vn= pVn−1+ Vn−2, V0= 2, V1= p.

With α, β = p ±pp2_{+ 4
/2, they admit the following expressions in the Binet forms}

Un= αn− βn α − β and Vn = α n_{+ β}n_. When α = 1+√5 2 (or equivalently q = (1 − √

5 )/(1 +√5 ) ), the sequence {Un} is reduced to the

Fibonacci sequence {Fn} and the sequence {Vn} is reduced to the Lucas sequence {Ln}.

For n ≥ k ≥ 1, we will use generalized Fibonomial coefficients n k  U = U1U2. . . Un (U1U2. . . Uk)(U1U2. . . Un−k) with n 0  U =n n  U = 1. When p = 1, they reduces to the usual Fibonomial coefficient, denoted byn

k

F. For more details

about the usual Fibonomial or generalized Fibonomial coefficients, their properties and interesting appearances in various places from number theory to linear algebra see [5, 6, 7, 8, 16, 17, 18].

Our approach will essentially be based on the following connection between the generalized Fibonomial and Gaussian q-binomial coefficients

n k  U = αk(n−k)n k  q

with q = β/α or, equivalently α = i/√q.

2010 Mathematics Subject Classification. Primary 11B39; Secondary 05A30.

Key words and phrases. Gaussian q-binomial coefficients, Fibonomial and Lucanomial Coefficients, Sums iden-tites, Partial fraction decomposition.

Furthermore, we will use generalized Lucanomial coefficients n k  V = V1V2. . . Vn (V1V2. . . Vk)(V1V2. . . Vn−k) withn n

V = 1 where Vn is the nth generalized Lucas number.

When Vn = Ln, the generalized Lucanomial coefficients are reduced to the Lucanomial

coeffi-cients denoted byn k L: n k  L = L1L2. . . Ln (L1L2. . . Lk)(L1L2. . . Ln−k) .

The link between the generalized Lucanomial and Gaussian q-binomial coefficients is n k  V = αk(n−k)n k  −q with q = −α−2 or α = i/√q.

There are many kind of sums including Gaussian q-binomial coefficients with certain weight functions or generalized Fibonomial coefficients with generalized Fibonacci and Lucas numbers as coefficients (for more details, see [9, 10, 11, 12, 13, 14, 15, 16, 18]).

Marques and Trojovsky [16] provide various sums including Fibonomial coefficients, Fibonacci and Lucas numbers. For example, for positive integers m and n, they show that

4m+2 X j=0 4m + 2 j  F (−1)j(j+1)2 _L 2m+1−j = − 4m + 2 4n + 3  F F4n+3 F2m+1 and 4m+2 X j=0 4m j  F (−1)j(j−1)2 _F n+4m−j = 1 2F2m+n 4m X j=0 4m j  F (−1)j(j−1)2 _L 2m−j.

Quite recently, Kılı¸c and Prodinger [11] compute three types of sums involving products of Gaussian q-binomial coefficients. They are of the following forms: for any real number a

n X k=0 n + k k  q n k  q (−1)kq−nk+(k2)(a − qk_), n X k=0 n + k k  q n k  q (−1)kq−nk+(k2) 1 q−k_{− a} and n X k=0 n + k − 1 k  q n k  q (−1)kq−nk+(k+12 ) a − q −k b − q−k.

Then they present interesting applications to generalized Fibonomial and Lucanomial sums. They prove their results by the partial fraction decomposition method. Recently the method has been explored in proving various functional identities, for more details we refer to [1, 2, 3, 4].

In this paper, we shall investigate sums of products of two Gaussian q-binomial coefficients, one of which includes two additional parameters, with a parametric rational weight function. Our results will generalize the results of [11], introducing two additional parameters. By means of partial fraction decomposition, we shall first prove the main theorems and then give two corollaries from each of them. Then these q-binomial identities will be transformed into Fibonomial sums as consequences.

We mainly compute five types of the q-binomial sums with certain weight functions: For any positive integer m, and any real numbers p, q, a and b,

n X k=0 m + k n  p,q n k  q (−1)kq(k+12 )−kn 1 1 − aqk, n X k=0 m + k n  p,q n k  q (−1)kq(k2)−kn  a + 1 q−k  , n X k=0 m + k − 1 n − 1  p,q n k  q (−1)kq(k2)−nk b − q −k a − q−k  , n X k=0 m − k n  p,q n k  q (−1)kq(k2) 1 q−k_{− a} and n X k=0 m − k n  p,q n k  q (−1)kq(k+12 ) 1 − aqrn−k+c
.

Note that when m = n and p = q in the above sums, three of them give us the results of [11]. By specifying the parameters, one could derive many different Fibonomial-Fibonacci-Lucas consequences.

2. The Main results

The results we will present throughout the paper will be satisfied for a, b, p, q, w ∈ R, nonnegative integer n and positive integer m. We start with our first result:

Theorem 2.1. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  p,q n k  q (−1)kq(k+12 )−kn 1 1 − aqk = a n(q; q)n (p; q)_n × a−1pqm−n_{; q}
n (a; q)n+1 . Proof. Consider the left-hand side of the claim

n X k=0 m + k n  p,q n k  q (−1)kq(k+12 )−kn 1 q−k_{− a} = n X k=0 (p; q)_m+k (p; q)_m−n+k(p; q)_n (q; q)_n (q; q)_n−k(q; q)_k(−1) k q(k+12 )−kn 1 q−k_{− a} = (q; q)n (p; q)_n n X k=0 (p; q)_m+k (p; q)_m−n+k 1 (q; q)_n−k(q; q)_k (−1) k q−12k(2n−k+1) 1 q−k_{− a} = (q; q)n (p; q)_n n X k=0 1 − pqm−n+k
. . . 1 − pqm+k−1
(q; q)_n−k(q; q)_k (−1) k q−12k(2n−k+1) 1 q−k_{− a}

or without the constant factors consider

n X k=0 qkn _q−k_{− pq}m−n
_{. . . q}−k_{− pq}m−1
(q; q)_n−k(q; q)_k (−1) k q−12k(2n−k+1) 1 q−k_{− a} = n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)_n−k(q; q)_k (−1) k q(k2) 1 q−k_{− a}.

Define the functions f (z) := (z − pq m−n_{) z − pq}m−n+1
_{. . . z − pq}m−1
(1 − z)(1 − zq) . . . (1 − zqn₎ 1 z − a and A (z) := (z − pq m−n_{) z − pq}m−n+1
_{. . . z − pq}m−1
(1 − z)(1 − zq) . . . (1 − zqn₎ .

Thus set f (z) = A (z)_z−a1 . Then the partial fraction expansion reads f (z) = n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k(z − q−k) (−1)kq(k2) 1 q−k_{− a}+ F (n, a) z − a . We multiply this by z and let z → ∞:

0 = n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k (−1)kq(k2) 1 q−k_{− a}+ F (n, a), where F (n, a) = −(z − pq m−n_{) z − pq}m−n+1
_{. . . z − pq}m−1
(1 − z)(1 − zq) . . . (1 − zqn_{)     z=a} = −(a − pq m−n_{) a − pq}m−n+1
_{. . . a − pq}m−1
(1 − a)(1 − aq) . . . (1 − aqn₎ = −a n _{1 − a}−1_pqm−n
1 − a−1pqm−n+1
_{. . . 1 − a}−1_pqm−1
(1 − a)(1 − aq) . . . (1 − aqn₎ = −an a −1_pqm−n_{; q}
n (a; q)n+1 . Thus we write n X k=0 q−k− pqm−n
q−k− pqm−n+1
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k (−1)kq(k2) 1 q−k_{− a} = an _a−1_pqm−n_{; q}
n (a; q)n+1 or n X k=0 (p; q)_m+k(q; q)_n (p; q)_n(p; q)_m−n+k(q)k(q)n−k (−1)kq(k2)qk−kn 1 1 − aqk = a n(q; q)n (p; q)_n × a−1pqm−n; q
_n (a; q)n+1 , which, after some rearrangements, gives us

n X k=0 m + k n  p,q n k  q (−1)kq(k+12 )−kn 1 1 − aqk = a n(q; q)n (p; q)_n × a−1pqm−n_{; q}
n (a; q)n+1 , as claimed. _

As consequences of Theorem 2.1, we give the following results: Corollary 2.1.1. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  −q n k  q (−1)kq(k2)−kn 1 1 + wqrn+c−k−1 = (−1)nq−(n+12 ) (q; q)n (−q; q)_n × wqm+(r−1)n+c; q
_n −wqn(r−1)+c−1_{; q}
n+1 .

Proof. It is enough to take p = −q and a = −w−1q−rn−c+1 in Theorem 2.1. Then the claimed

result follows after some rearrangements. _

Corollary 2.1.2. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  q n k  q (−1)kq(k2)−kn 1 1 − wqrn−k+c = (−1)n q −(n+1 2 ) 1 − wq(r−1)n+c rn + c n −1 wq,q m + rn + c n  wq,q .

Proof. It is enough to take p = q and a = w−1q−rn−c in Theorem 2.1. Then the claimed result follows after some rearrangements with the definitions of the Gaussian q-binomial coefficient and

the q-Pochhammer notation. _

Theorem 2.2. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  p,q n k  q (−1)kq(k2)−kn  a + 1 q−k  = (−1)nq−(n+12 ) [a + pn_qmn_](q; q)n (p; q)_n. Proof. Consider the left-hand side of the claim

n X k=0 m + k n  p,q n k  q (−1)kq(k2)−kn  a + 1 q−k  = (q; q)n (p; q)_n n X k=0 (p; q)_m+k (p; q)_m−n+k 1 (q; q)_n−k(q; q)_k (−1) k q(k2)−kn  a + 1 q−k  = (q; q)n (p; q)_n n X k=0 1 − pqm−n+k
_{. . . 1 − pq}m+k−1
(q; q)_n−k(q; q)_k (−1) k q(k2)−kn  a + 1 q−k  = (q; q)n (p; q)_n n X k=0 qkn _q−k_{− pq}m−n
_{. . . q}−k_{− pq}m−1
(q; q)_n−k(q; q)_k (−1) k q(k2)−kn  a + 1 q−k  = (q; q)n (p; q)_n n X k=0 q−k_{− pq}m−n
_{. . . q}−k_{− pq}m−1
(q; q)_n−k(q; q)_k (−1) k q(k2)  a + 1 q−k 

or without the constant it takes the form

n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)_n−k(q; q)_k (−1) k q(k2)  a + 1 q−k  . Now define the functions

f (z) := (z − pq m−n_{) z − pq}m−n+1
_{. . . z − pq}m−1
(1 − z)(1 − zq) . . . (1 − zqn₎  a +1 z  and A (z) := (z − pq m−n_{) z − pq}m−n+1
_{. . . z − pq}m−1
(1 − z)(1 − zq) . . . (1 − zqn₎ .

Thus we see that f (z) = A (z) a +1_z
. Then the partial fraction expansion reads f (z) = n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k (−1)kq(k2)  a + 1 q−k  +C z.

We multiply this by z and let z → ∞: a(−1)nq−(n+12 ) = n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k (−1)kq(k2)  a + 1 q−k  + C, where C = −(−1)npnq(m−n)n+(n2). Thus we write n X k=0 q−k− pqm−n
_{. . . q}−k_{− pq}m−1
(q; q)k(q; q)n−k (−1)kq(k2)  a + 1 q−k  = a(−1)nq−(n+12 ) + (−1)npnq(m−n)n+( n 2) or n X k=0 1 − pqm−n+k
. . . 1 − pqm+k−1
(q; q)k(q; q)n−k (−1)kq(k2)−kn  a + 1 q−k  = (−1)nhaq−(n+12 ) + pnq(m−n)n+( n 2) i or n X k=0 m + k n  p,q n k  q (−1)kq(k2)−kn  a + 1 q−k  = (−1)nq−(n+12 ) (q; q)n (p; q)_n[a + p n_qmn_{] ,} as claimed. _

As consequences of Theorem 2.2, by taking special values of p and a, we give the following corollaries:

Corollary 2.2.1. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  q n k  q (−1)kq−12k(2n−k+1) _{1 + wq}rn+c+k
= (−1)n_q−( n+1 2 )  1 + wqc+n(1+m+r). Corollary 2.2.2. For nonnegative integers m and n such that m ≥ n,

n X k=0 m + k n  −q n k  q 1 − wqk+rn+c
(−1)k q(k2)−kn = (−1)nq−(n+12 ) h 1 − (−1)nwqn(m+r+1)+ci (q; q)n (−q; q)_n. Theorem 2.3. For positive integers m and n such that m ≥ n,

n X k=0 m + k − 1 n − 1  p,q n k  q (−1)kq(k2)−nkq −k_{− b} q−k_{− a} = a n−1_{(a − b)} (q; q)n (p; q)_n−1 × (qm−n_{p/a; q)} n−1 (a; q)n+1 . Proof. Rewrite the LHS of the claim as

n X k=0 m + k − 1 n − 1  p,q n k  q (−1)kq(k2)−nkq −k_{− b} q−k_{− a} = n X k=0 (p; q)_m+k−1 (p; q)_n−1(p; q)_m−n+k (q; q)_n (q; q)k(q; q)n−k (−1)kq(k2)−nkq −k_{− b} q−k_{− a}

= (q; q)n (p; q)_n−1 n X k=0 (p; q)_m+k−1 (p; q)_m−n+k 1 (q; q)k(q; q)n−k (−1)kq(k2)−nkq −k_{− b} q−k_{− a} = (q; q)n (p; q)_n−1 n X k=0 1 − pqm−n+k
_{. . . (1 − pq}m+k−2₎ (q; q)k(q; q)n−k (−1)kq(k2)−nkq −k_{− b} q−k_{− a}.

Without constant factors, consider

n X k=0 qkn _q−k_{− pq}m−n
_{. . . (q}−k_{− pq}m−2₎ (q)k(q)n−k (−1)kq(k2)−nkq −k_{− b} q−k_{− a} = n X k=0 q−k− pqm−n
_{. . . (q}−k_{− pq}m−2₎ (q)k(q)n−k (−1)kq(k2) q −k_{− b} q−k_{− a}.

Now define the function

A (z) := (z − pq

m−n_{) . . . (z − pq}m−2₎

(1 − z)(1 − zq) . . . (1 − zqn₎

z − b z − a. The partial fraction decomposition of A(z) takes the form:

A(z) = n X k=0 q−k− pqm−n
_{. . . (q}−k_{− pq}m−2₎ (q; q)k(q; q)n−k (−1)kq(k+12 ) q −k_{− b} q−k_{− a} 1 1 − zqk + F (n, a) z − a . Now we multiply this relation by z and let z → ∞ and obtain

0 = lim z→∞  n X k=0 q−k_{− pq}m−n
_{. . . (q}−k_{− pq}m−2₎ (q; q)k(q; q)n−k (−1)kq(k+12 ) q −k_{− b} q−k_{− a} z (1 − zqk₎+ F (n, a) z z − a  , which gives us 0 = n X k=0 q−k− pqm−n
_{. . . (q}−k_{− pq}m−2₎ (q; q)k(q; q)n−k (−1)k−1q(k2) q −k_{− b} q−k_{− a}+ F (n, a) or n X k=0 q−k− pqm−n
_{. . . (q}−k_{− pq}m−2₎ (q; q)k(q; q)n−k (−1)kq(k2) q −k_{− b} q−k_{− a}= F (n, a), where F (n, a) =(z − b)(z − pq m−n_{) z − pq}m−n+1
_{. . . (z − pq}m−2₎ (1 − z)(1 − zq) . . . (1 − zqn₎   _z=a = (a − b)(a − pq m−n_{) a − pq}m−n+1
_{. . . (a − pq}m−2₎ (1 − a)(1 − aq) . . . (1 − aqn₎ = a(−1)nq−(n+12 ) + an−1_{(a − b)} (qm−n_{p/a; q)} n−2 (a; q)n+1 . Thus we get n X k=0 q−k− pqm−n
_{. . . (q}−k_{− pq}m−2₎ (q; q)k(q; q)n−k (−1)kq(k2) q −k_{− b} q−k_{− a}= a n−1_{(a − b)}(q m−n_{p/a; q)} n−2 (a; q)n+1 or n X k=0 q−k(n−1) 1 − pqm−n+k
_{. . . (1 − pq}m+k−2₎ (q; q)k(q; q)n−k (−1)kq(k2) q −k_{− b} q−k_{− a} = a n−1_{(a − b)}(q m−n_{p/a; q)} n−2 (a; q)n+1 or

(q; q)_n (p; q)_n−1 n X k=0 1 − pqm−n+k
_{. . . (1 − pq}m+k−2₎ (q; q)k(q; q)n−k (−1)kq−(n−1)k+(k2) q −k_{− b} q−k_{− a} = an−1(a − b)(q m−n_{p/a; q)} n−2 (a; q)n+1 or n X k=0 m + k − 1 n − 1  p,q n k  q (−1)kq−(n−1)k+(k2) q −k_{− b} q−k_{− a} = a n−1_{(a − b)} (q; q)n (p; q)_n−1 × (qm−n_{p/a; q)} n−1 (a; q)n+1 , as wanted. _

As consequences of Theorem 2.3, we give the following corollaries by taking special values of p and a:

Corollary 2.3.1. For positive integers m and n such that m ≥ n,

n X k=0 m + k − 1 n − 1  q n k  q (−1)kq(k2)−nk+k 1 − q k 1 − qn+k = −q n(n−1)_{(1 − q}n₎2 q m−2n+1_{; q}
n−1 (qn_{; q)} n+1 . Corollary 2.3.2. For positive integers m and n such that m ≥ n,

n X k=0 m + k − 1 n − 1  −q n k  q (−1)kq(k2)−nk1 − q n+k 1 + qk = (1 − q n₎ (q; q)n (−q; q)_n−1 × qm−n+1; q
_n−1 (−1; q)n+1 . Finally we give the following theorems without proof which could be similarly done. Theorem 2.4. For nonnegative integers m and n such that m ≥ 2n,

n X k=0 m − k n  p,q n k  q (−1)kq(k2) 1 q−k_{− a} = (q; q)n (p; q)_n × (apqm−n_{; q)} n (a; q)n+1 . As consequences of Theorem 2.4, we give the following corollaries:

Corollary 2.4.1. For nonnegative integers m and n such that m ≥ 2n,

n X k=0 m − k n  q n k  q (−1)kq(k+12 ) 1 1 − wqk+rn+c−1 = 1 1 − wq(r+1)n+c−1 rn + m + c n  w,q (r + 1) n + c − 1 n −1 w,q . Corollary 2.4.2. For nonnegative integers m and n such that m ≥ 2n,

n X k=0 m − k n  −q n k  q (−1)kq(k+12 ) 1 1 + wqk+rn+c = (q; q)_n (−q; q)_n × wq(r−1)n+m+c+1_{; q}
n (−wqrn+c_{; q)} n+1 . Now we give our main last result.

Theorem 2.5. For nonnegative integers m and n such that m ≥ 2n,

n X k=0 m − k n  p,q n k  q (−1)kq(k+12 ) a + q−k
= h a + pnqn(m−n−1)i(q; q)n (p; q)_n. As consequences of Theorem 2.5, we have the following corollaries:

Corollary 2.5.1. For nonnegative integers m and n such that m ≥ 2n, n X k=0 m − k n  −q n k  q (−1)kq(k+12 ) 1 − (−1)n_wqrn−k+c
= h 1 − wq(m−n+r)n+ci (q; q)n (−q; q)_n. Corollary 2.5.2. For nonnegative integers m and n such that m ≥ 2n,

n X k=0 m − k n  q n k  q (−1)kq(k+12 ) 1 − wqrn−k+c
= h 1 − wq(m−n+r)n+ci. 3. Applications

In this section, we shall give applications of our results as the generalized Fibonomial sums identities.

(1) For positive even n, positive m and all integers r and c such that m − (r + 1) n ≥ c and rn + c ≥ 1, n X k=0 m + k n  U n k  U (−1)( k 2) 1 Uk+rn+c = 1 U(r+1)n+c m − rn − c n  U (r + 1) n + c − 1 n −1 U , n X k=0 m + k n  U n k  U (−1)(k2) 1 Vk+rn+c = 1 V(r+1)n+c m − rn − c n  V (r + 1) n + c − 1 n −1 V . (2) For positive even n, positive m, and all integers r and c such that rn + c ≥ 1,

n X k=0 m + k n  V n k  U (−1)(k2) 1 Uk+rn+c = 1 U(r+1)n+c (r + 1) n + c − 1 n −1 U m − rn − c n  V , n X k=0 m + k n  V n k  U (−1)(k2) 1 Vk+rn+c = ∆n 1 V(r+1)n+c n Y k=1 Uk Vk !2 (r + 1) n + c − 1 n −1 V m − rn − c n  V . (3) For all positive odd n, positive m and all integers r and c such that m − (r + 1) n ≥ c and

(r − 1) n + c ≥ 1, n X k=0 m + k n  U n k  U (−1)(k2) U−1 rn−k+c= (−1) n−1 2 1 U(r−1)n+c rn + m + c n  U rn + c n −1 U , n X k=0 m + k n  U n k  U (−1)( k 2) V−1 rn−k+c= (−1) n−1 2 1 V(r−1)n+c rn + m + c n  V rn + c n −1 V . (4) For positive odd n, positive m and all integers r and c such that m − (r + 1) n ≥ c and

(r − 1)n + c ≥ 1, n X k=0 m + k n  V n k  U (−1)(k2) 1 Urn−k+c = (−1)(n−1)/2 1 U(r−1)n+c rn + c n −1 U m + rn + c n  V , n X k=0 m + k n  V n k  U (−1)(k2) 1 Vrn−k+c = (−1)n−12 _∆n n Y k=1 Uk Vk !2 rn + m + c n  U rn + c n −1 V .

(5) For positive n, m and all integers r and c such that m − (r + 1) n + c ≥ 1 and rn + c ≥ 1, n X k=0 m − k n  U n k  U (−1)(k2) U−1 k+rn+c= 1 U(r+1)n+c m + rn + c n  U (r + 1) n + c − 1 n −1 U , n X k=0 m − k n  U n k  U (−1)( k 2) V−1 k+rn+c= 1 V(r+1)n+c m + rn + c n  V (r + 1) n + c − 1 n −1 V .

(6) For positive n, m and all integers r and c such that m − (r + 1) n + c ≥ 1 and rn + c ≥ 1,

n X k=0 m − k n  U n k  U (−1)( k 2) 1 Uk+rn+c = 1 Urn+c (r + 1) n + c n −1 U m + rn + c n  V , n X k=0 m − k n  V n k  U (−1)(k2) 1 Vk+rn+c = ∆n n Y k=1 Uk Vk !2 m + rn + c n  U (r + 1) n + c n −1 V .

(7) For odd n, positive m, and, all integers r and c

n X k=0 m + k n  U n k  U (−1)(k2) U k+rn+c= (−1) n−1 2 _U n(m+r+1)+c, n X k=0 m + k n  U n k  U (−1)(k2) V k+rn+c= (−1) n−1 2 _V n(m+r+1)+c and n X k=0 m + k n  V n k  U (−1)(k2) U k+rn+c= (−1) n−1 2 _∆n−12 V_n(m+r+1)+c n Y k=1 Uk Vk , n X k=0 m + k n  V n k  U (−1)(k2) V k+rn+c= (−1) n−1 2 _∆n+12 U_n(m+r+1)+c n Y k=1 Uk Vk .

(8) For nonnegative integers n, m, and all integers r and c such that m ≥ n

n X k=0 m − k n  U n k  U (−1)(k2) V rn−k+c= V(m−n+r)n+c n X k=0 m − k n  U n k  U (−1)(k2) U rn−k+c= U(m−n+r)n+c

(9) For nonnegative integers n, m, and all integers r and c such that m ≥ n

n X k=0 m − k n  V n k  U (−1)( k 2) U rn−k+c= ∆b n 2c  n Y k=1 Uk Vk ( U(m−n+r)n+c if n is even, V(m−n+r)n+c if n is odd, and n X k=0 m − k n  U n k  U (−1)(k2) V rn−k+c= V(m−n+r)n+c, n X k=0 m − k n  U n k  U (−1)(k2) U rn−k+c= U(m−n+r)n+c.

As a showcase, we will show how the above generalized Fibonomial-Lucanomial-Fibonacci or Lucas corollaries are obtained from our main results. For this, we will prove the first corollary, that is, n X k=0 m + k n  U n k  U (−1)(k2) 1 Uk+rn+c = 1 U(r+1)n+c m − rn − c n  U (r + 1) n + c − 1 n −1 U

for positive even n. First we convert it into q-form by using the relationship given in the Introduc-tion. Thus, after rearranging and simplifications, the claim takes the form

n X k=0 αn(m+k−n)m + k n  q αk(n−k)n k  q (−1)( k 2) 1 − q αk+rn+c−1_{(1 − q}k+rn+c₎ = 1 − q α(r+1)n+c−1 _{1 − q}(r+1)n+c
× αn(m−rn−c−n)m − rn − c n  q α−n((r+1)n+c−1−n)(r + 1) n + c − 1 n −1 q or n X k=0 m + k n  q n k  q (−1)(k2) 1 (1 − qk+rn+c₎α −k+2kn−k2 = α−2n(c+nr) 1 1 − q(r+1)n+c
m − rn − c n  q (r + 1) n + c − 1 n −1 q . For the case of even n, we have to prove that

n X k=0 m + k n  q n k  q 1 (1 − qk+rn+c₎(−1) k q(k+12 )−kn = qn(c+nr) 1 1 − q(r+1)n+c
m − rn − c n  q (r + 1) n + c − 1 n −1 q . If we take a = qrn+cand p = q in Theorem 1, we write

n X k=0 m + k n  q n k  q (−1)kq(k+12 )−kn 1 1 − qk+rn+c = q n(rn+c)(q−rn−cqqm−n; q)n (qrn+c_{; q)} n+1 , which, after some rearrangement, is equal to

qn(rn+c) q 1−c+m−(r+1)n_{; q}
n (qrn+c_{; q)} n+1 = qn(rn+c) (q; q)m−rn−c (q; q)_n(r+1)+c (q; q)_rn+c−1 (q; q)_{m−(r+1)n−c} = qn(rn+c) 1 1 − qn(r+1)+c (q; q)_m−rn−c (q; q)_{m−(r+1)n−c}(q; q)_n (q; q)_rn+c−1(q; q)_n (q; q)_n(r+1)+c−1 = qn(c+nr) 1 1 − q(r+1)n+c
m − rn − c n  q (r + 1) n + c − 1 n −1 q , as expected. So the claim is true.

References

[1] CHEN, X.—CHU, W.: Further3F2(4₃) series via Gould–Hsu inversions, Integral Trans. Special Functions,

24(6) (2013), 441–469.

[2] CHU, W.: A binomial coefficient identity associated with Beukers’ conjecture on Ap´ery numbers. Electron. J. Combin., 11(1) (2004), #N15, 3 pp.

[3] CHU, W.: Harmonic number identities and Hermite–Pad´e approximations to the logarithm function, J. Ap-prox. Theory, 137(1) (2005), 42–56.

[4] CHU, W.: Partial fraction decompositions and trigonometric sum identities, Proc. Amer. Math. Soc. 136(1) (2008), 229–237.

[5] GOULD, H. W.: The bracket function and Founten´e–Ward generalized binomial coefficients with application to fibonomial coefficients, The Fibonacci Quarterly 7 (1969), 23–40.

[6] HOGGATT JR., V. E.: Fibonacci numbers and generalized binomial coefficients, The Fibonacci Quarterly 5 (1967), 383–400.

[7] HORADAM, A. F.: Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J. 32 (1965), 437–446.

[8] KILIC¸ , E.: The generalized Fibonomial matrix, European J. Comb. 29(3) (2008), 701–711.

[9] KILIC¸ , E.: Evaluation of sums containing triple aerated generalized Fibonomial coefficients, Math. Slovaca 67(2) (2017), 355–370.

[10] KILIC¸ , E.—OHTSUKA, H.—AKKUS¸, I: Some generalized Fibonomial sums related with the Gaussian q-binomial sums, Bull. Math. Soc. Sci. Math. Roumanie 55(1) (103) (2012), 51–61.

[11] KILIC¸ , E.—PRODINGER, H.: Evaluation of sums involving products of Gaussian q-binomial coefficients with applications to Fibonomial sums, Turkish J. Math. 41(3) (2017), 707–716.

[12] KILIC¸ , E.—PRODINGER, H.—AKKUS, I.—OHTSUKA, H.: Formulas for Fibonomial sums with generalized Fibonacci and Lucas coefficients, The Fibonacci Quarterly 49(4) (2011), 320–329.

[13] KILIC¸ , E.—PRODINGER, H.: Evaluation of sums involving Gaussian q-binomial coefficients with rational weight functions, Int. J. Number Theory 12(2) (2016), 495–504.

[14] KILIC¸ , E.—PRODINGER, H.: Closed form evaluation of sums containing squares of Fibonomial coefficients, Math. Slovaca 66(3) (2016), 757–767.

[15] LI, N. N.—CHU, W.: q-Derivative operator proof for a conjecture of Melham, Discrete Appl. Math. 177 (2014), 158–164.

[16] MARQUES, D.—TROJOVSKY, P.: On some new sums of Fibonomial coefficients, The Fibonacci Quarterly 50(2) (2012), 155–162.

[17] SEIBERT, J.—TROJOVSKY, P.: On some identities for the Fibonomial coefficients, Math. Slovaca 55 (2005), 9–19.

[18] TROJOVSKY, P.: On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math. 155(15) (2007), 2017–2024.

* Department of Mathematics, TOBB University of Economics and Technology, 06560 Ankara, Turkey

E-mail address: ekilic@etu.edu.tr

** Department of Mathematics, University of Stellenbosch, 7602 Stellenbosch, South Africa E-mail address: hproding@sun.ac.za