This paper deals with matrix transformations that preserve the (p,q)-convexity of sequences. The main result gives the necessary and sufficient conditions for a nonnegative infinite matrix A to preserve the (p,q)-convexity of sequences. Further, we give examples of such matrices for different values of p and q.

1. Introduction

If p>0, q>0, then the sequence {xn} of real numbers is said to be (p,q)-convex if(1)Δp,qxn=xn-p+qxn-1+pqxn-2≥0for n≥2. The operator Δp,q generates the second-order difference Δ2 when p=q=1. Several authors [1–3] have proved various results on the convex sequences defined by Δ2xn≥0. Other authors [4, 5] have studied the classes of sequences satisfying Δ1,q(xn)≥0. Also, the necessary and sufficient conditions for a sequence to be a (p,q)-convex sequence can be found in [6]. Moreover, some inequalities on (p,q)-convex sequences are given in [7, 8].

In [9–11], the authors discuss the matrix transformations that preserve (p,q)-convexity of sequences in the case of a lower triangular matrix with a particular type of matrix transformation. But the question of a general infinite matrix preserving (p,q)-convexity has not been considered anywhere in the literature. This paper deals with the necessary and sufficient conditions for a nonnegative infinite matrix to preserve (p,q)-convexity in both settings when p≠q and p=q.

2. Preliminaries

For any given sequence {xn}, we can find a corresponding sequence {ck} such that(2)c0=x0,c1=x1-p+qc0and, for k≥2,(3)ck=xk-∑i=0k-1pk-i+pk-i-1q+⋯+pqk-i-1+qk-ici,which implies that {xn} can be represented by(4)x0=c0,x1=c1+p+qc0, and, for n≥2,(5)xn=cn+p+qcn-1+p2+pq+q2cn-2+⋯+pn+pn-1q+⋯+pqn-1+qnc0=cn+∑i=1npi+pi-1q+⋯+pqn-i+qicn-i.As a consequence, we get the following lemma. A variation of this lemma can be found in [6].

Lemma 1.

If the sequence {xn} is given by representation (5), then Δp,q(xn)=cn. Thus, the sequence {xn} is (p,q)-convex if and only if cn≥0 for n≥2.

Proof.

It suffices to show that Δp,q(xn)=xn-(p+q)xn-1+pqxn-2=cn for n≥2. Using (5), (6)Δp,qxn=cn+p+qcn-1+p2+pq+q2cn-2+⋯+∑0npn-kqkc0-p+qcn-1+p+qcn-2+p2+pq+q2cn-3+⋯+∑0n-1pn-k-1qkc0+pqcn-2+p+qcn-3+p2+pq+q2cn-4+⋯+∑0n-2pn-k-2qkc0.On the right side, we see that the coefficient of cn=1, and the coefficient of cn-r=0 for r=1,2,…,n. Thus,(7)Δp,qxn=cnfor n≥2.Hence, we have the previous lemma.

Also, in (5), the representation of xn in terms of cn can be written as follows:(8)xn=cn+∑i=0n-1pn-i+pn-i-1q+⋯+qn-ici=cn+∑i=0n-1pn-i+1-qn-i+1p-qci,if p≠qcn+∑i=0n-1n-i+1pn-ici,if p=q=∑i=0npn-i+1-qn-i+1p-qci,if p≠q∑i=0nn-i+1pn-iciif p=q.Now, we give below some definitions. Let A=[an,k] be a nonnegative infinite matrix defining a sequence to sequence transformation by(9)Axn=∑k=0∞an,kxk.Then, we define the matrices [αn,k] and [βn,k] as(10)αn,k=∑j=k∞pj-kan,j=an,k+pan,k+1+p2an,k+2+⋯,βn,i=∑k=i∞qk-iαn,k=αn,i+qαn,i+1+q2αn,i+2+⋯=∑k=i∞qk-i∑j=k∞pj-kan,j.Interchanging the order of summation, we get, for each n=0,1,2,…, and i=0,1,2,…,(11)βn,i=∑j=i∞∑k=ijqk-ipj-kan,j=∑j=i∞pj-i+qpj-i-1+q2pj-i-2+⋯+qj-ian,j=∑j=i∞pj-i+1-qj-i+1p-qan,j,if p≠q∑j=i∞j-i+1pj-ian,j,if p=q.Furthermore, for n≥2,(12)Δp,qβn,i=βn,i-p+qβn-1,i+pqβn-2,i=∑j=i∞pj-i+1-qj-i+1p-qΔp,qan,j,if p≠q∑j=i∞j-i+1pj-iΔp,qan,j,if p=q. In order for the matrix [βn,i] to be well-defined, we need the matrix [an,k] to satisfy certain conditions which will depend on the values of p and q.

(I) When p≠q, due to symmetry of p and q in the definition of βn,i, it is sufficient to consider the following cases:(13)a0<p,q<1b0<p<1,q=1cp>1,q=1dp>1,0<q<1ep,q>1

Case (a). For 0<p,q<1, we require the matrix A to satisfy that, for each n,(14)∑k=1∞kan,k<∞. Thus, using (11) and p,q<1, we have (15)βn,i=∑j=i∞pj-i+qpj-i-1+⋯+qj-ian,j<∑j=i∞j-i+1an,j=∑j=i∞j-ian,j+∑j=i∞an,j<∞by 14.Thus, βn,i is well-defined.

Case (b). For 0<p<1,q=1, we require the matrix A to satisfy that, for each n,(16)∑k=0∞an,k<∞.Then using (11), we have (17)βn,i=∑j=i∞1-pj-i+11-pan,j=11-p1-pan,i+1-p2an,i+1+⋯<11-pan,i+an,i+1+⋯, since 0<p<1<∞by 16.Thus, βn,i is well-defined.

For the cases (c), (d), and (e), we require the matrix A to satisfy that, for each n,(18)∑k=0∞pkan,k<∞.

Case (c). When p>1,q=1, we have, as in the case (b), (19)βn,i=∑j=i∞pj-i+1-1p-1an,j=pp-1∑j=i∞pj-i-1pan,j<pp-1∑j=i∞pj-ian,j≤pp-1∑j=i∞pjan,j<∞by 18. Thus, βn,i is well-defined.

Case (d). When p>1,0<q<1, from (11),(20)βn,i=1p-q∑j=i∞pj-i+1-qj-i+1an,j. Since q<p, using (18), we have ∑j=i∞qj-ian,j<∑j=i∞pj-ian,j<∞. Therefore, (21)βn,i=pp-q∑j=i∞pj-ian,j-qp-q∑j=i∞qj-ian,j<∞. Thus βn,i is well-defined.

Case (e). When p,q>1, we can assume without loss of generality that p>q.

Proceeding as in case (d), we see that βn,i is well-defined in this case also.

(II) When p=q, we consider the following cases:(22)f0<p<1gp=1hp>1.

Case (f). For 0<p<1, we require the matrix A to satisfy that, for each n,(23)∑k=1∞kan,k<∞. Then, using (11), we have (24)βn,i=∑j=i∞j-i+1pj-ian,j<∑j=i∞j-i+1an,j=∑j=i∞j-ian,j+∑j=i∞an,j<∞by 23. Thus, βn,i is well-defined.

Case (g). When p=1, Δp,q-convexity reduces to the well-known second-order convexity Δ2, which has been investigated in detail in [3].

Case (h). For p>1, we require the matrix A to satisfy that, for each n,(25)∑k=1∞kpkan,k<∞. Then, using (11), we have (26)βn,i=∑j=i∞j-i+1pj-ian,j≤∑j=i∞j-ipj-ian,j+∑j=i∞pjan,j<∞by 25.Thus, βn,i is well-defined.

3. Main Results

In this section, we prove the necessary and sufficient conditions for a nonnegative infinite matrix A to transform a (p,q)-convex sequence into a (p,q)-convex sequence showing that each column of the corresponding matrix [βn,k] is a (p,q)-convex sequence.

First, we consider the values of p and q, where p≠q results in the cases listed in (13).

Theorem 2.

For p≠q, a nonnegative infinite matrix A satisfying (14), (16), or (18), corresponding to the cases listed in (13), preserves (p,q)-convexity of sequences if and only if, for n=2,3,4,…,

Δp,qβn,0=0

Δp,qβn,1=0

Δp,qβn,i≥0 for i≥2

where the matrix [βn,i] is defined by(27)βn,i=∑j=i∞pj-i+1-qj-i+1p-qan,j.Proof.

First, we prove a result on the transformed sequence of any (p,q)-convex sequence {xn}. Now, we have, from (8),(28)xn=∑i=0npn-i+1-qn-i+1p-qci,where ci≥0 for i≥2 by Lemma 1. Then, the nth term of the transformed sequence is (29)Axn=∑k=0∞an,kxk=∑k=0∞an,k∑i=0kpk-i+1-qk-i+1p-qci. Interchanging the order of summation, (30)Axn=∑i=0∞ci∑k=i∞pk-i+1-qk-i+1p-qan,k=c0∑k=0∞pk+1-qk+1p-qan,k+c1∑k=1∞pk-qkp-qan,k+∑i=2∞ci∑k=i∞pk-i+1-qk-i+1p-qan,k. From (11), we have(31)Axn=c0βn,0+c1βn,1+∑i=2∞ciβn,i.Then, for n≥2, (32)Δp,qAxn=Axn-p+qAxn-1+pqAxn-2=c0βn,0+c1βn,1+∑i=2∞ciβn,i-p+qc0βn-1,0+c1βn-1,1+∑i=2∞ciβn-1,i+pqc0βn-2,0+c1βn-2,1+∑i=2∞ciβn-2,i=c0βn,0-p+qβn-1,0+pqβn-2,0+c1βn,1-p+qβn-1,1+pqβn-2,1+∑i=2∞ciβn,i-p+qβn-1,i+pqβn-2,i. Thus, for any (p,q)-convex sequence {xn},(33)Δp,qAxn=c0Δp,qβn,0+c1Δp,qβn,1+∑i=2∞ciΔp,qβn,i. Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then, by (33),(34)Δp,qAxn≥0. Thus, the sequence (Ax)n is also (p,q)-convex.

Conversely, assume that the matrix A preserves (p,q)-convexity of the sequences. Suppose that the condition (i) fails to hold. Then there exists an integer N≥2 such that(35)Δp,qβN,0=L≠0. Consider the following sequence:(36)u=-L,-p2-q2p-qL,-p3-q3p-qL,…. Then {un} is a (p,q)-convex sequence because, using (2) and Lemma 1,(37)c0=u0=-L,c1=u1-p+qc0=0 and, for i≥2,(38)ci=Δp,qui=ui-p+qui-1+pqui-2=-pi+1-qi+1p-qL+p+qpi-qip-qL-pqpi-1-qi-1p-qL=0. Thus, from (33), for the transformed sequence {(Au)n}, (39)Δp,qAuN=c0Δp,qβN,0+c1Δp,qβN,1+∑i=2∞ciΔp,qβN,i=-L2<0,which contradicts that the transformed sequence {(Au)n} must be (p,q)-convex.

Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: (40)v=0,-L,-p2-q2p-qL,-p3-q3p-qL,…, which implies that(41)c0=0,c1=-L,ci=0for i≥2.Now, suppose that the condition (iii) is not true. Then there exists an integer j≥2 such that the jth column-sequence of the matrix [βn,k] is not (p,q)-convex. That is, for some N≥2,(42)Δp,qβN,j=L<0. Now, consider the following sequence: (43)x=0,…,0,1,p2-q2p-q,p3-q3p-q,….↓x0↓xj-1↓xj↓xj+1Then, {xn} is a (p,q)-convex sequence, because, using (2) and Lemma 1, we get(44)ci=0for 0≤i≤j-1;cj=1;cj+1=xj+1-p+qxj+pqxj-1=0;and, for i≥j+2,(45)ci=Δp,qxi=0as in 38. But, from (33), (46)Δp,qAxN=c0Δp,qβN,0+c1Δp,qβN,1+∑i=2∞ciΔp,qβN,i=cjΔp,qβN,j=L<0,which again contradicts that {Ax} is a (p,q)-convex sequence. This completes the proof.

Theorem 2 generalizes the necessary and sufficient conditions given in [9, Theorem 2, p. 8] in the case of p=1 and q>0 with q≠1.

Next, we consider the values of p and q where p=q results in the cases listed in (22).

Theorem 3.

For p=q, a nonnegative infinite matrix A satisfying (23) or (25), corresponding to the cases listed in (22), preserves (p,q)-convexity of sequences if and only if, for n=2,3,4,…,

Δp,p(βn,0)=0

Δp,p(βn,1)=0

Δp,p(βn,i)≥0 for i=2,3,…,

where the matrix [βn,i] is defined by(47)βn,i=∑j=i∞j-i+1pj-ian,j.Proof.

First we prove a result on the transformed sequence of any (p,p)-convex sequence {xn}. Now, we have, from (8),(48)xn=∑i=0nn-i+1pn-ici, where ci≥0 for i≥2 by Lemma 1. Then, the nth term of the transformed sequence is (49)Axn=∑k=0∞an,kxk=∑k=0∞an,k∑i=0kk-i+1pk-ici. Interchanging the order of summation, (50)Axn=∑i=0∞ci∑k=i∞k-i+1pk-ian,k=c0∑k=0∞k+1pkan,k+c1∑k=1∞kpk-1an,k+∑i=2∞ci∑k=i∞k-i+1pk-ian,k. From (11), we have(51)Axn=c0βn,0+c1βn,1+∑i=2∞ciβn,i. Then, for n≥2, (52)Δp,pAxn=Axn-2pAxn-1+p2Axn-2=c0βn,0+c1βn,1+∑i=2∞ciβn,i-2pc0βn-1,0+c1βn-1,1+∑i=2∞ciβn-1,i+p2c0βn-2,0+c1βn-2,1+∑i=2∞ciβn-2,i.Thus, for any (p,p)-convex sequence {xn},(53)Δp,pAxn=c0Δp,pβn,0+c1Δp,pβn,1+∑i=2∞ciΔp,pβn,i. Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then by (53),(54)Δp,pAxn≥0. Thus, the sequence (Ax)n is also (p,p)-convex.

Conversely, assume that the matrix A preserves (p,p)-convexity of sequences.

Suppose that the condition (i) fails to hold. Then there exists an integer N≥2 such that(55)Δp,pβN,0=L≠0. Consider the following sequence:(56)u=-L,-2pL,-3p2L,…. It is easy to see, using (2) and Lemma 1, that u is a (p,p)-convex sequence with(57)c0=uo=-L,ci=0for i≥1. Thus, from (53), for the transformed sequence {(Au)n}, (58)Δp,pAuN=c0Δp,pβN,0+c1Δp,pβN,1+∑i=2∞ciΔp,pβN,i=-L2<0, which contradicts that {(Au)n} must be (p,p)-convex.

Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: (59)v=0,-L,-2pL,-3p2L,…, which implies that(60)c0=0,c1=-L,ci=0for i≥2. Now, suppose that the condition (iii) is not true. Then there exists an integer j≥2 such that the jth column-sequence of the matrix [βn,k] is not (p,p)-convex. That is, for some N≥2,(61)Δp,pβN,j=L<0. Consider the (p,p)-convex sequence:(62)x=0,…,0,1,2p,3p2,….↓x0↓xj-1↓xj↓xj+1 We see that, as in the proof of Theorem 2,(63)Δp,pAxN=L<0,which contradicts that {Ax} is a (p,p)-convex sequence.

We see that the result on the convexity of sequences given in [3, p. 331] is a particular case of Theorem 3 when p=q=1. Also, this theorem generalizes the necessary and sufficient conditions for a triangular matrix given in [9, p. 4].

4. Examples

We give below examples of (p,q)-convexity preserving matrices for each of the cases (a) through (h) given in (13) and (22).

Example for Case (a). Considering 0<p,q<1, and p≠q, we can assume, without loss of generality, that p<q. Let the matrix A=[an,k] be defined by (64)an,k=pn,if k=0,pnqkk,if k≥1.Then, for each n,(65)∑k=1∞kan,k=∑k=1∞pnqk=pnq1-q<∞. Thus, by (14), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 2 because, for n≥2, using (12),(66)Δp,qβn,i=∑j=i∞pj-i+1-qj-i+1p-qΔp,qan,j, in which (67)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=pn-p+qpn-1+pqpn-2,if j=0,qjjpn-p+qpn-1+pqpn-2,if j≥1=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (b). Considering 0<p<1, q=1, let the matrix A=[an,k] be defined by(68)an,k=pk. Then, for each n,(69)∑k=0∞an,k=∑k=0∞pk=11-p<∞.Thus, by (16), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 2 because, for n≥2, using (12),(70)Δp,1βn,i=∑j=i∞pj-i+1-1p-1Δp,1an,j, in which(71)Δp,1an,j=an,j-p+1an-1,j+pan-2,j=pj-p+1pj+pj+1=0. Therefore, the matrix A preserves (p,1)-convexity of sequences.

Example for Case (c). Considering p>1,q=1, let matrix A=[an,k] be defined by(72)an,k=1p2k. Then, for each n,(73)∑k=0∞pkan,k=∑k=0∞1pk=11-1/p<∞. Thus, by (18), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 2 because, for n≥2, as in the previous example (b),(74)Δp,1an,j=an,j-p+1an-1,j+pan-2,j=1p2j-p+11p2j+p1p2j=0. Therefore, the matrix A preserves (p,1)-convexity of sequences.

Example for Case (d). Considering p>1,0<q<1, let matrix A=[an,k] be defined by(75)an,k=pn-2k. Then, for each n,(76)∑k=0∞pkan,k=∑k=0∞pn-k=pn+1p-1<∞. Thus, by (18), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 2 because, for n≥2, using (12),(77)Δp,qβn,i=∑j=i∞pj-i+1-qj-i+1p-qΔp,qan,j, in which(78)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=1p2jpn-p+qpn-1+pqpn-2=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (e). Considering p,q>1 and p≠q, we can assume, without loss of generality, that p>q. Let the matrix A=[an,k] be defined by(79)an,k=pn-2kqk.Then, for each n,(80)∑k=0∞pkan,k=pn∑k=0∞qpk=pn+1p-q<∞. Thus, by (18), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 2 because, for n≥2, as in the previous example (d),(81)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=qjp2jpn-p+qpn-1+pqpn-2=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (f). Considering 0<p=q<1, let the matrix A=[an,k] be defined by (82)an,k=pn,if k=0,pn+kk,if k≥1. Then, for each n,(83)∑k=1∞kan,k=∑k=1∞pn+k=pn∑k=1∞pk=pn+111-p<∞. Thus, by (23), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 3 because, for n≥2, using (12),(84)Δp,pβn,i=∑j=i∞j-i+1pj-iΔp,pan,j, in which (85)Δp,pan,j=an,j-2pan-1,j+p2an-2,j=pn-2ppn-1+p2pn-2,if j=0,pjjpn-2ppn-1+p2pn-2,if j≥1=0. Therefore, the matrix A preserves (p,p)-convexity of sequences.

Examples for Case (g). They can be found in [3], since Δ1,1 is the same as the second-order convexity Δ2.

Example for Case (h). Considering p=q>1, let the matrix A=[an,k] be defined by (86)an,k=pnn+2,if k=0,pn-2kn+2k,if k≥1.Therefore, for each n, (87)∑k=1∞kpkan,k=∑k=1∞pn-kn+2=pnn+2∑k=1∞1pk=n+2pnp-1<∞. Thus, by (23), βn,i is well-defined for n=0,1,2,… and i=0,1,2,…. The matrix A satisfies the three conditions of Theorem 3 because, for n≥2, using (12),(88)Δp,pβn,i=∑j=i∞j-i+1pj-iΔp,pan,j, in which (89)Δp,pan,j=an,j-2pan-1,j+p2an-2,j=pnn+2-2pnn+1+pnnif j=0,n+2pn-2jj-2n+1pn-2jj+npn-2jj,if j≥1.=0. Therefore, the matrix A preserves the convexity of sequences.

We conclude this paper by giving an example of an infinite matrix which does not preserve (p,q)-convexity of sequences.

It is interesting to notice that the Borel matrix preserves the (1,1)-convexity of sequences [3, p. 336], but it does not preserve (p,p)-convexity when p≠1.

The Borel matrix B=[bn,k] is defined by(90)bn,k=nkenk!. Then, for each n,(91)∑k=1∞kbn,k=nen∑k=1∞nk-1k-1!=n<∞,(92)∑k=1∞kpkbn,k=npen∑k=1∞npk-1k-1!=npenenp<∞.Thus, for each of the cases, 0<p<1 and p>1, we see that (23) and (25) are satisfied and hence βn,i is well-defined for n=0,1,2,… and i=0,1,2….

From (11),(93)βn,i=∑j=i∞j-i+1pj-ibn,j.Therefore, (94)βn,0=∑j=0∞j+1pjnjenj!=1en∑j=0∞jpnjj!+∑j=0∞pnjj!=1enpnepn+epn=enp-1pn+1, which implies that (95)Δp,pβn,0=βn,0-2pβn-1,0+p2βn-2,0=enp-1pn+1-2pen-1p-1pn-1+1+p2en-2p-1pn-2+1=enp-1e2ppn+1ep-pe2+2p2eep-pe>0, since ep-pe>0 when p≠1. Thus, the condition (i) of Theorem 3 fails in the case of Borel matrix.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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