Nonnegative Infinite Matrices that Preserve ( p , q )-Convexity of Sequences

and Applied Analysis 3 Thus, using (11) and p, q < 1, we have βn,i = ∞ ∑ j=i (pj−i + qpj−i−1 + ⋅ ⋅ ⋅ + qj−i) an,j < ∞ ∑ j=i (j − i + 1) an,j = ∞ ∑ j=i (j − i) an,j + ∞ ∑ j=i an,j < ∞

In [9][10][11], the authors discuss the matrix transformations that preserve (, )-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 (, )-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 (, )-convexity in both settings when  ̸ =  and  = .

Preliminaries
For any given sequence {  }, we can find a corresponding sequence {  } such that and, for  ≥ 2, which implies that {  } can be represented by and, for  ≥ 2, As a consequence, we get the following lemma.A variation of this lemma can be found in [6].
On the right side, we see that the coefficient of   = 1, and the coefficient of  − = 0 for  = 1, 2, . . ., .Thus, Hence, we have the previous lemma.Also, in (5), the representation of   in terms of   can be written as follows: Now, we give below some definitions.Let  = [ , ] be a nonnegative infinite matrix defining a sequence to sequence transformation by Then, we define the matrices [ , ] and [ , ] as Interchanging the order of summation, we get, for each  = 0, 1, 2, . .., and  = 0, 1, 2, . .., Furthermore, for  ≥ 2, In order for the matrix [ , ] to be well-defined, we need the matrix [ , ] to satisfy certain conditions which will depend on the values of  and .
(I) When  ̸ = , due to symmetry of  and  in the definition of  , , it is sufficient to consider the following cases: Case (a).For 0 < ,  < 1, we require the matrix  to satisfy that, for each , Thus, using (11) and ,  < 1, we have Thus,  , is well-defined.
For the cases (c), (d), and (e), we require the matrix  to satisfy that, for each , Case (c).When  > 1,  = 1, we have, as in the case (b), Thus,  , is well-defined.
Case (e).When ,  > 1, we can assume without loss of generality that  > .
Proceeding as in case (d), we see that  , is well-defined in this case also.
(II) When  = , we consider the following cases: Case (f).For 0 <  < 1, we require the matrix  to satisfy that, for each , Then, using (11), we have Thus,  , is well-defined.

Case (h).
For  > 1, we require the matrix  to satisfy that, for each , Then, using (11), we have Thus,  , is well-defined.

Main Results
In this section, we prove the necessary and sufficient conditions for a nonnegative infinite matrix  to transform a (, )convex sequence into a (, )-convex sequence showing that each column of the corresponding matrix [ , ] is a (, )convex sequence.First, we consider the values of  and , where  ̸ =  results in the cases listed in (13).
Next, suppose that the condition (ii) is not true.This case can be settled by a similar argument by considering the following sequence: which implies that Now, suppose that the condition (iii) is not true.Then there exists an integer  ≥ 2 such that the th column-sequence of the matrix [ , ] is not (, )-convex.That is, for some  ≥ 2, Δ , ( , ) =  < 0.