Special number sequences such as the Fibonacci, Lucas, Horadam, Jacobsthal, and balancing numbers sequences are widely studied in number theory. Finding generalizations of such number sequences, establishing new identities, and finding applications of these sequences in other branches of mathematics have become very a popular research goal; see, for example, [6, 13]. Mikkawy and Sogabe [2] introduced a new family of k-Fibonacci numbers Fn(k) where n is of kind sk+r, 0≤r<k. Among other properties, then gave a relation to the classic Fibonacci numbers. Later, Özkan et al. [8] further studied this sequence and introduced a new family of k-Lucas numbers. Kumari et al. [7] extended the study to Mersenne numbers and investigated some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials.

Motivated by these works on new families of sequences, in this paper, we study a new family of k-balancing and k-Lucas balancing numbers and associated polynomials, where the concept of balancing numbers and balancers was originally introduced in 1999 by Behera and Panda [1].

A natural number n is said to be a balancing number [1] with balancer r if it satisfies the Diophantine equation

The balancing numbers B_n and Lucas balancing numbers C_n are defined as

The first few terms of balancing and Lucas-balancing sequence are

Closed form formulas play an important roll in establishing many algebraic identities. The closed form formulas for balancing and Lucas-balancing numbers [11], are given as

where λ1=3+8 and λ2=3−8 are the roots of the characteristic equation x2−6x+1=0.

We will use the following useful relations for the balancing and Lucas-balancing numbers.

Lemma 1.1. ([12]) For all integers m and n, we have

• 1. 2BmCm=B2m.

• 2. Bm+n+Bm−n=2BmCn.

• 3. Bm+n−Bm−n=2CmBn.

• 4. Bm−nBm+n=Bm2−Bn2.

• 5. Cn−mCn+m−Cn2=12(C2m−1).

• 6. C2n=2Cn2−1.

• 7. Cn2=8Bn2+1.

Fibonacci numbers have many generalizations, in which both the initial values and/or the recurrence relation are modified. The k-Fibonacci numbers, tribonacci numbers, Horadam numbers, generalized Fibonacci and Leonardo numbers, higher order Fibonacci numbers, are some examples of generalization of Fibonacci numbers. Likewise, k-balancing numbers {Bk,n} and k-Lucas balancing numbers {Ck,n}, both generalisations of balancing numbers, were introduced and studied by Özkoc and Tekcan [9] and Ray [11]. These sequences are given by the following recurrences:

Later in [10], Ray extended the k-balancing numbers Bk,n to the sequence of balancing polynomials {Bn(x)} by replacing k with a real variable x and presented numerous properties of balancing polynomials. Frontczak [3] also studied the balancing polynomials by relating them to Chebyshev polynomials.

In this paper, we give a new generalization of balancing and Lucas-balancing numbers à la [2], which we call the generalized k-balancing and k-Lucas balancing numbers. They are defined in Section 2. Then in Section 3, we give associated polynomials having a connection with balancing polynomials.

Our main defintion is as follows.

Definition 2.1. Let k∈ℕ and n∈ℕ∪{0} then ∃!s,r∈ℕ∪{0} such that n=sk+r, 0≤r<k. The generalized k-balancing and k-Lucas balancing numbers Bn(k) and Cn(k) are defined as

where λ1 and λ2 are the roots of the characteristic equation corresponding to balancing sequence (1.1).

From Definition 2.1 and Eqn. (1.3), one gets the following relation between the generalized k-balancing and k-Lucas balancing numbers and the balancing and Lucas balancing numbers.

If k=1 then r=0 and hence n=s. Therefore, Bn(1) and Cn(1) are the classic balancing and Lucas balancing numbers i.e. Bn(1)=Bn and Cn(1)=Cn.

In the case that k=2 or 3 we note some identities showing the relations between generalized k-balancing numbers and balancing numbers:

• 1. B2s(2)=Bs2.

• 2. B2s+1(2)=BsBs+1.

• 3. B3s(3)=Bs3.

• 4. B3s+1(3)=Bs2Bs+1.

• 5. B3s+2(3)=BsBs+12.

• 1. C2s(2)=Cs2=(C2s+1)/2.

• 2. C2s+1(2)=CsCs+1.

• 3. C3s(3)=Cs3.

• 4. C3s+1(3)=Cs2Cs+1.

• 5. C3s+2(3)=CsCs+12.

One can also check that B2s+1(2)=6B2s(2)−B2s−1(2) and B3s+1(3)=6B3s(3)−B3s−1(3). The same recurrences hold for generalized k-Lucas balancing numbers.

For k=1,2,3,4,5, a list of first few numbers of the generalized k-balancing and k-Lucas balancing numbers are displayed in Table 1 and 2.

Table 1 . Generalized k-balancing numbers.

n↓	Bn(k)	k=1	k=2	k=3	k=4	k=5
0	B0(k)	0	0	0	0	0
1	B1(k)	1	0	0	0	0
2	B2(k)	6	1	0	0	0
3	B3(k)	35	6	1	0	0
4	B4(k)	204	36	6	1	0
5	B5(k)	1189	210	36	6	1
6	B6(k)	6930	1225	216	36	6
7	B7(k)	40391	7140	1260	216	36

Table 2 . Generalized k-Lucas balancing numbers.

n↓	Cn(k)	k=1	k=2	k=3	k=4	k=5
0	C0(k)	1	1	1	1	1
1	C1(k)	3	3	3	3	3
2	C2(k)	17	9	9	9	9
3	C3(k)	99	51	27	27	27
4	C4(k)	577	289	153	81	81
5	C5(k)	3363	1683	867	459	243
6	C6(k)	19601	9801	4913	2601	1377
7	C7(k)	114243	57123	28611	14739	7803

Theorem 2.2. For k,s∈ℕ, we have

Proof. If r=0 then n=sk and hence from Eqn. (2.2) the above results are proved.

Lemma 2.3. We have

Proof. Since,

Similarly, the second result holds.

Thus, we conclude that

A similar identity holds for Csk+r(k).

Theorem 2.4. For a≥1 and n∈ℕ such that n=sk+r, 0≤r<k, we have

Proof. From Eqn. (2.2) for Wi=Bi or Wi=Ci, we can write

Thus using Theorem 2.2 in the above equation, the result holds.

In particular for a=2 in Theorem 2.4, we have

Theorem 2.5. For k,s∈ℕ such that n=sk+1, the following relations are verified:

Proof. For the first identity, from Theorem 2.2 and Lemma 2.3, we write

A similar argument holds for the second identity.

Theorem 2.6. Let s,k∈ℕ, then for fixed k,s, the following results hold:

Proof. For the first identity (2.4), using relation (2.2), we write

Similarly, for the second identity (2.4), we have

For the third identity (2.5), since from (2.2) we write Bsk+a(k)=Bsk−aBs+1a=Bsk(Bs+1/Bs)a. Thus

And, for the last identity (2.6), note that ∑ a=1kaxa−1=(1−kxk−1+(k−1)xk)/(1−x)2. Thus,

Note that Theorem 2.6 is also valid for the generalized k-Lucas balancing numbers {Csk}.

Theorem 2.7. For k,s∈ℕ, we have

Proof. Results follow from Eqn. (2.2).

Theorem 2.8 (Cassini's identity). For s,k≥2, we have

Proof. For a≠1, from Eqn. (2.2) we write

and if a=1 then with Eqn. (2.2) and Lemma 2.3

A similar argument holds for the second identity.

Theorem 2.9. For integers s, s₁, s₂ and k≥1, we have

• 1. B2sk(k)=2kBsk(k)Csk(k).

• 2. (Bs1+s2+Bs1−s2)k=2kBs1k(k)Cs2k(k).

• 3. (Bs1+s2−Bs1−s2)k=2kCs1k(k)Bs2k(k).

• 4. B2s1(2)−B2s2(2)=Bs1+s2Bs1−s2.

• 5. C2sk(k)=(2Cs2−1)k=(2Bs2+1)k.

• 6. (Cs1+s2+Cs1−s2)k=2kCs1k(k)Cs2k(k).

• 7. (Cs1+s2−Cs1−s2)k=16kBs1k(k)Bs2k(k).

• 8. C2s1(2)−C2s2(2)=8Bs1+s2Bs1−s2.

Proof. From Theorem 2.2 and 1 of Lemma 1.1, note that for the first identity, we have

For the second identity, from 2 of Lemma 1.1, we have

For the third identity, from 3 of Lemma 1.1, we have

For the fourth identity, from Theorem 2.2 and 4 of Lemma 1.1, we write

The argument for identities 5 - 8 are similar to that of 1 - 4 using Theorem 2.2 and Lemma 1.1.

For n≥0, the balancing and Lucas balancing polynomials Bn(x) and Cn(x) satisfy the recurrence relation

but with the initial values as B0(x)=0, B1(x)=1 and C0(x)=1, C1(x)=3x, respectively. The Binet type formulas for these polynomials are, respectively,

where λ1=(3x+9x2−1)/2 and λ2=(3x−9x2−1)/2 are roots of the characteristic equation λ2−6xλ+1=0.

Now, we define the generalized k-balancing and k-Lucas balancing polynomials in a similar fashion to the preceding section.

Definition 3.1. Let k∈ℕ and for n≥0, ∃! s,r ∈ \mathbb{N}∪ {0} such that n=sk+r, 0≤r<k. Then the generalized k-balancing and k-Lucas balancing polynomials Bn(k)(x) and Cn(k)(x) are defined as

From Binet's formula (3.2) and Definition 3.1, we deduce the following relations between newly introduced sequences and existing one

For the case k=1, we get r=0. Hence, from Eqn. (3.3), we have Ws(1)(x)=Ws(x).

For instance at k=2,3 in (3.3), we have noted some identities showing relations between newly introduced polynomials sequences and classic balancing/Lucas balancing polynomials:

• 1. W2s(2)(x)=Ws2(x).

• 2. W2s+1(2)(x)=Ws(x)Ws+1(x).

• 3. W3s(3)(x)=Ws3(x).

• 4. W3s+1(3)(x)=Ws2(x)Ws+1(x).

• 5. W3s+2(3)(x)=Ws(x)Ws+12(x).

• 6. W4s(4)(x)=Ws4(x).

• 7. W4s+1(4)(x)=Ws3(x)Ws+1(x).

• 8. W4s+2(4)(x)=Ws2(x)Ws+12(x).

• 9. W4s+3(4)(x)=Ws(x)Ws+13(x).

Also the recurrence relations W2s+1(2)(x)=6W2s(2)(x)−W2s−1(2)(x) for k=2 and W3s+1(3)(x)=6W3s(3)(x)−W3s−1(3)(x) for k=3 are verified.

Theorem 3.2. For k,s∈ℕ, we have

Proof. If r=0 then sk+r= sk and hence from Eqn. (3.3) the result follows immediately.

By a similar argument to Lemma 2.3, we have Wsk−1(k)(x)=Ws−1(x)Wsk−1(x) which will be used in the next theorem.

Theorem 3.3. For k,s∈ℕ such that n=sk+1, the following recurrence relation is satisfied.

Proof. From Eqn. (3.3) and Eqn. (3.1), we have

Theorem 3.4. We have

Proof. Since,

An analogue argument to the preceding section proves the following theorems, so we omit the proofs.

Theorem 3.5. For k,s∈ℕ we have,

Theorem 3.6 (Cassini's identity) Let k,s ≥ 2, we have

Theorem 3.7. For integers s, s₁, s₂ and k ≥ 1, the following relations are valid:

• 1. B2sk(k)(x)=Bsk(k)(x)[Bs+1(x)−Bs−1(x)]k.

• 2. B2s1(2)(x)−B2s2(2)(x)=Bs1−s2(x)Bs1+s2(x).

• 3. [Bs+1(x)−Bs−1(x)]k=2kCsk(k)(x).

• 4. [3xBs(x)−Bs−1(x)]k=2kCsk(k)(x).

• 5. [Bs+1(x)−3xBs(x)]k=Csk(k)(x).

• 6. [Bs+12(x)−Bs−12(x)]k=6kxkB2sk(k)(x).

• 7. [Cs+12(x)−Cs−12(x)]k=6kxk(9x2−1)kB2sk(k)(x).

• 8. C2s1(2)(x)+C2s2(2)(x)=Cs1−s2(x)Cs1+s2(x)+1.

• 9. C2s(2)(x)−(9x2−1)B2s(2)(x)=1.

• 10.[3xCs−1(x)+(9x2−1)Bs−1(x)]k=Csk(k)(x).

• 11. B2sk(k)(x)=2kBsk(k)(x)Csk(k)(x).

• 12. C2sk(k)(x)=[2Cs2(x)−1]k.

Proof. The arguments for these identities are analog to the proof of Theorem 2.9 and can be easily verified using Proposition 2.3 of [3] and Section 3 of [10] along with Theorem 3.2.