Sequences of positive integers have long been studied and many special integer sequences are known to have applications in different areas of science. Many researchers devote their attention to special sequences, such a Pell, Pell-Lucas, and Modified Pell sequences, which satisfy a second-order recurrence relation. Horadam studied various properties of Pell numbers and Pell polynomials. Ercolano found generating matrices for Pell sequences. Many mathematicians have looked at generalizations of Pell sequences one gets by adding one or two parameters to the recursion relation but not altering the initial conditions. Identities and generalting functions for the k-Pell numbers were established in [3]. The authors of [2] investigated (s,t)-Pell and (s,t)-Pell-Lucas sequences and their matrix representations. In [4], (s,t)-Pell and Pell-Lucas numbers are studied using matrix methods. In [5], the exponential representations of the Jacobsthal matrix sequences were found. In this paper we give the definitions of Pell sequence and its parametrized generalizations. Using the elements of the sequence, we establish matrix sequences for the integer sequences. We demonstrate the exponential matrices for the Pell matrix sequence and its generalizations by various methods.

As seen [1, 6], the recurrence relation with initial conditions for the Pell sequence is given as

pn=2pn-1+pn-2,p0=0 and p1=1, n≥2.

The characteristic equation for the recurrence relation of the Pell sequence is

x2-2x-1=0

with roots r1=1+2 and r2=1-2. It is easily seen that r1+r2=2,r1r2=-1 and r1-r2=22. The Binet formula for showing Pell numbers as a function of the roots r1, r2 is established as

(1.1)pn=r1n-r2nr1-r2.

The sequence can be generalized using one parameter k, which is any positive integer. The k-Pell sequence pk,nn∈N in [3] is demonstrated by

pk,n=2pk,n-1+kpk,n-2,pk,0=0 and pk,1=1,n≥2.

It has the characteristic equation

x2-2x-k=0

with roots

rk,1=1+1+k and rk,2=1-1+k.

So, the following properties are established

(1.2)rk,1rk,2=-k, rk,1+rk,2=2, rk,1-rk,2=21+k.

The Binet formula for the k-Pell sequence with roots rk,1 and rk,2 is given by

(1.3)pk,n=rk,1n-rk,2nrk,1-rk,2.

As established in [4, 5], the two-parameter Pell sequence (s,t) -Pell sequence pn(s,t)n∈N is obtained by the following recurrence relation

pns,t=2spn-1s,t+tpn-2s,t,p0(s,t)=0,p1(s,t)=1,n≥2,

where s,t are real numbers such that s>0,t≠0 and s2+t>0. The characteristic equation of this recurrence relation is

x2-2sx-t=0

with roots r1(s,t)=s+s2+t and r2(s,t)=s-s2+t. The roots satisfy the relations:

(1.4)r1(s,t)r2(s,t)=-t, r1(s,t)+r2(s,t)=2s , r1(s,t)- r2(s,t)=2s2+t.

The Binet formula for (s,t)-Pell numbers with the roots r1(s,t), r2(s,t) is given by

(1.5)pns,t= r1n(s,t)- r2n(s,t) r1(s,t)- r2(s,t).

The Pell matrix sequence Pnn∈N is defined in [2] by the recurrence relation

(2.1)Pn+1=2Pn+Pn-1, P0=1001,P1=2110.

The elements of Pell matrix sequence are the elements of Pell sequence such that

Pn=pn+1pnpnpn-1.

The k-Pell matrix sequence Pk,nn∈N is established by

(2.2)Pk,n+1=2Pk,n+kPk,n-1,Pk,0=1001, Pk,1=21k0.

The elements of k-Pell matrix sequence are the elements of k-Pell sequence such that

Pk,n=pk,n+1pk,nkpk,nkpk,n-1.

The s,t-Pell matrix sequence Pn(s,t)n∈N is defined in [4, 5] by

(2.3)Pn+1s,t=2sPns,t+tPn-1s,t,P0s,t=1001,P1s,t=2s1t0.

The elements of (s,t)-Pell matrix sequence are the elements of (s,t)-Pell sequence such that

Pns,t=pn+1s,tpns,ttpns,ttpn-1s,t.

Lemma 2.1. Assume s,t are real numbers such that s,t>0,t≠0, n≥1 an integer, k any positive integer, the following identities hold:

Pn=P1n, Pk,n=Pk,1n, Pns,t=P1ns,t.

Proof. The proof is made by induction method. We want to prove the last equality that Pns,t=P1ns,t. For n=1, it is easily seen that the assumption is true. Assume that Pks,t=P1ks,t is true for k≤n. We want to seek for the assumption is valid for k=n+1:

P1n+1s,t=P1ns,tP1s,t=Pns,tP1s,t =pn+1s,tpns,ttpns,ttpn-1s,t2s1t0 =2spn+1s,t+tpns,tpn+1s,ttpn+1s,ttpns,t =Pn+1s,t.

If we choose s=t=1 in this equality, we get the first equality.

Similarly, If we choose s=1, t=k, we get the second equality.

Lemma 2.2. Assume s,t are real numbers such that s,t>0,t≠0 and n≥1 an integer, k any positive integer, the following identities hold:

Pm+n=PmPn, Pk,m+n=Pk,mPk,n, Pm+ns,t=Pms,tPns,t.

Proof. The proof is made by induction method. We want to prove the second equality that Pk,m+n=Pk,mPk,n. For n=0, it is easily seen that the assumption is true. Assume that Pk,m+i=Pk,mPk,i is true for i≤n. We want to seek for the assumption is valid for i=n+1:

Pk,m+n+1=2Pk,m+n+kPk,m+n-1=2Pk,mPk,n+kPk,mPk,n-1 =Pk,m(2Pk,n+kPk,n-1)=Pk,mPk,n+1.

The other proofs are made by using the same procedure.

In this section, we want to present the exponential representations of the nth element of Pell matrix sequence and the nth element of generalized Pell matrix sequences. If a function f(z) of a complex variable z has a Maclaurin series expansion f(z)=∑∞k=0akzk which converges for z≺R, then the matrix series ∑∞k=0akAk converges, provided A is square and each of its eigenvalues has absolute value less than R. In such a case, f(A) is defined as f(A)=∑∞k=0akAk .

Theorem 3.1. For any integer n≥0, the exponential representation of the nth element of (s,t)-Pell matrix sequence is in the following form:

ePn(s,t)=-(-t)n∑∞k=0pnk-n(s,t)pn(s,t)k!I2+∑∞k=0pnk(s,t)pn(s,t)k!P1n(s,t)

where I2 is the identity matrix and pn(s,t) is defined in [4]. The theorem gives us the opportunity finding the exponential representations of the nth element of (s,t)-Pell matrix sequence using the nth power of first element of (s,t)-Pell matrix sequence. Similarly, the exponential representation of the nth element of k-Pell matrix sequence is

ePk,n=-(-k)n∑∞i=0pk,ni-npk,ni!I2+∑∞i=0pk,nipk,ni!Pk,1n,

and the exponential representation of the nth element of the Pell matrix sequence is

ePn=∑∞k=0pnkpnk!P1n-(-1)n∑∞k=0pnk-npnk!I2.

Proof. The eigenvalues of P1(s,t)=2st10 are r1(s,t)=s+s2+t and r2(s,t)=s-s2+t. By Lemma 2.1, we know that Pns,t=P1ns,t. Therefore, the eigenvalues of Pn(s,t) are r1n(s,t)=s+s2+tn and r2n(s,t)=s-s2+tn. By the equality ePn(s,t)=a1Pn(s,t)+a0I2, we get

er1n(s,t)=a1r1n(s,t)+a0,er2n(s,t)=a1r2n(s,t)+a0.

By these equations, the values of a0, a1 arefound. If we substitute the values of a0, a1, it is obtained that

ePn(s,t)=r1n(s,t)er2n(s,t)-r2n(s,t)er1n(s,t)r1n(s,t)-r2n(s,t)I2+er1n(s,t)-er2n(s,t)r1n(s,t)-r2n(s,t)Pn(s,t).

Applying the Maclaurin series expansion of expx and (1.4), (1.5), we obtain the following:

er1n(s,t)-er2n(s,t)r1n(s,t)-r2n(s,t)=∑∞k=0r1nk(s,t)-r2nk(s,t)r1n(s,t)-r2n(s,t)1k!=∑∞k=0pnk(s,t)pn(s,t)k!,
r1n(s,t)er2n(s,t)=r1n(s,t)∑∞k=0r2nk(s,t)1k!=(-t)n∑∞k=0r2nk-n(s,t)1k!,

and

r2n(s,t)er1n(s,t)=r2n(s,t)∑∞k=0r1nk(s,t)1k!=(-t)n∑∞k=0r1nk-n(s,t)1k!.

If the results are combined

ePn(s,t)=r1n(s,t)er2n(s,t)-r2n(s,t)er1n(s,t)r1n(s,t)-r2n(s,t)I2+er1n(s,t)-er2n(s,t)r1n(s,t)-r2n(s,t)Pn(s,t) =(-t)n∑∞k=0r2nk-n(s,t)1k!-(-t)n∑∞k=0r1nk-n(s,t)1k!I2/r1n(s,t)-r2n(s,t) +∑∞k=0pnk(s,t)pn(s,t)k!Pn(s,t) =-(-t)n∑∞k=0pnk-n(s,t)pn(s,t)k!I2+∑∞k=0pnk(s,t)pn(s,t)k!P1n(s,t).

If we choose s=t=1, we can apply this result for classic Pell matrix sequence defined in (2.1). The eigenvalues of P1=2110 matrix are r1=1+2 and r2=1-2. By r1r2=-1

ePn=r1ner2n-r2ner1nr1n-r2nI2+er1n-er2nr1n-r2nPn =∑∞k=0pnkpnk!P1n-(-1)n∑∞k=0pnk-npnk!I2

where Pn is the nth element of the classic Pell matrix sequence.

If we choose s=1, t=k, we can apply this result for k-Pell matrix sequence defined in (2.2). The eigenvalues of Pk,1=21k0 matrix are 1+1+k and 1-1+k. By Lemma 2.1, the eigenvalues of Pk,n matrix as rk,1n =1+1+knand rk,2n=1-1+kn. By the Binet formula for Pk,n and rk,1rk,2=-k

ePk,n=rk,1nerk,2n-rk,2nerk,1nrk,1n-rk,2nI2+erk,1n-erk,2nrk,1n-rk,2nPk,n =(-k)n∑∞i=0rk,2ni-n1i!-(-k)n∑∞i=0rk,1ni-n1i!I2/rk,1n-rk,2n+∑∞i=0pk,nipk,ni!Pk,n =-(-k)n∑∞i=0pk,ni-npk,ni!I2+∑∞i=0pk,nipk,ni!Pk,1n

where Pk,n is the nth element of the k-Pell matrix sequence.

The following theorem shows us a second way for expressing the exponential representation of the nth element of (s,t)-Pell matrix sequence.

Theorem 3.2. For any integer n≥0, the exponential representation of the nth element of (s,t)-Pell matrix sequence is in the following form:

ePn(s,t)=Uer1n(s,t)00er2n(s,t)U-1

where U is an invertible matrix and

er1n(s,t)00er2n(s,t)=∑∞i=0tpni-1(s,t)i!I2-∑∞i=0pni(s,t)i!r1(s,t)00r2(s,t).

This result has more advantage for finding the exponential representations of the nth element of (s,t)-Pell matrix sequence. Because we only need the elements of the sequence (pn(s,t)).

Proof. Because of the eigenvalues of Pn(s,t) are r1(s,t)n and r2(s,t)n, there is an invertible U matrix such that

Pn(s,t)=Ur1(s,t)n00r2(s,t)nU-1.

Therefore, the exponential form is

ePn(s,t)=Uer1(s,t)n00er2(s,t)nU-1.

Then, we obtain

er1(s,t)n00er2(s,t)n=r1(s,t)er2(s,t)n-r2(s,t)er1(s,t)nr1(s,t)-r2(s,t)I2 +er2(s,t)n-er1(s,t)nr1(s,t)-r2(s,t)r1(s,t)00r2(s,t) =∑∞i=0(-t)r2ni-1(s,t)-r1ni-1(s,t)r1(s,t)-r2(s,t)1i!I2 +∑∞i=0r2ni(s,t)-r1ni(s,t)r1(s,t)-r2(s,t)1i!r1(s,t)00r2(s,t) =∑∞i=0tpni-1(s,t)i!I2-∑∞i=0pni(s,t)i!r1(s,t)00r2(s,t).

If we choose s=t=1, we can apply this result for classic Pell matrix sequence as

ePn=Uer1n00er2nU-1

where

er1n00er2n=r1er2n-r2er1nr1-r2I2+er1n-er2nr1-r2r100r2 =∑∞i=0r1ni-1-r2ni-1r1-r21i!I2+∑∞i=0r1ni-r2nir1-r21i!r100r2 =∑∞i=0pni-1i!I2+∑∞i=0pnii!r100r2.

If we choose s=1, t=k, we can apply this result for k-Pell matrix sequence. The eigenvalues of matrix are 1+1+k and 1-1+k. By Lemma 2.1, the eigenvalues of Pk,n matrix as rk,1n =1+1+kn and rk,2n=1-1+kn.By the Binet formula for Pk,n and rk,1rk,2=-k, we get

ePk,n=Uerk,1n00erk,2nU-1

where

erk,1n00erk,2n=rk,1erk,2n-rk,2erk,1nrk,1-rk,2I2+erk,1n-erk,2nrk,1-rk,2rk,100rk,2 =∑∞i=0krk,1ni-1-rk,2ni-1rk,1-rk,21i!I2+∑∞i=0rk,1ni-rk,2nirk,1-rk,21i!rk,100rk,2 =k∑∞i=0pk,ni-1i!I2+∑∞i=0pk,nii!1+1+k001-1+k.

Theorem 3.3. For n≥0, the exponential representation of the (2n)th element of Pell matrix sequence is given in the following form:

eP2n=p1+∑∞k=1p2nk-1k!I2+∑∞k=0p2nkk!r100r2.

Proof. By using Lemma 2.2, there is an invertible U matrix such that

P2n=PnPn=Ur12n00r22nU-1.

By using the properties of the exponential matrix, we have

eP2n=Uer12n00er22nU-1.

We also obtain

er12n00er22n=r1er22n-r2er12nr1-r2I2+er22n-er12nr1-r2r100r2,
er12n-er22nr1-r2=∑∞k=0r12nk-r22nkr1-r21k!=∑∞k=0p2nkk!,

and

r1er22n-r2er12nr1-r2=∑∞i=0r12ni-1-r22ni-1r1-r21i!=∑∞i=0p2ni-1i!.

By combining the results, the proof is completed.

Exponential representation of (2n)th k-Pell and (s,t)-Pell matrix sequences can be obtained by using the same procedure. We give the results as

ePk,2n=k∑∞i=1pk,2ni-1i!I2+∑∞i=0pk,2nii!rk,100rk,2,
eP2n(s,t)=t∑∞k=1p2nk-1(s,t)k!I2+∑∞k=0p2nk(s,t)k!r1(s,t)00r2(s,t).

Theorem 3.4. For n≥0, the exponential representation of the (2n)th element of (s,t)-Pell matrix sequence is computed in the following form:

eP2n(s,t)=-t2n∑∞k=0p2nk-2n(s,t)p2n(s,t)k!I2+∑∞k=0p2nk(s,t)p2n(s,t)k!P12n(s,t)

Proof. The eigenvalues of P1(s,t) are r1(s,t)=s+s2+t and r2(s,t)=s-s2+t. The eigenvalues of P2n(s,t) are r12n(s,t)=s+s2+t2n and r22n(s,t)=s-s2+t2n. By the equality eP2n(s,t)=a1P2n(s,t)+a0I2, we get

er12n(s,t)=a1r12n(s,t)+a0,er22n(s,t)=a1r22n(s,t)+a0.

By these equations, the values of a0, a1 arefound. If we substitute the values of a0, a1, it is obtained that

eP2n(s,t)=r12n(s,t)er22n(s,t)-r22n(s,t)er12n(s,t)r12n(s,t)-r22n(s,t)I2 +er12n(s,t)-er22n(s,t)r12n(s,t)-r22n(s,t)P2n(s,t) =∑∞k=01k!r22nk-2n(s,t)-r1nk-n(s,t)I2r12n(s,t)-r22n(s,t) +∑∞k=0p2nk(s,t)p2n(s,t)k!P2n(s,t) =-t2n∑∞k=0p2nk-2n(s,t)p2n(s,t)k!I2+∑∞k=0p2nk(s,t)p2n(s,t)k!P12n(s,t)

If we choose s=t=1, we can apply this result for the classic Pell matrix sequence

ePn=∑∞k=0p2nkp2nk!P12n-∑∞k=0p2nk-2np2nk!I2.

If we choose s=1, t=k, we can apply this result for k-Pell matrix sequence

ePk,n=-k2n∑∞i=0pk,2ni-2npk,2ni!I2+∑∞i=0pk,2nipk,2ni!Pk,12n.