# On double Hausdorff summability method

## Abstract

Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savaş and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savaş and Şevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices.

MSC:40F05, 40G05.

## Introduction and background

The basic theory of Hausdorff transformations for double sequences was developed by Adams  in 1933. Later a few authors studied double Hausdorff matrices; see e.g. Ramanujan  and Ustina .

Several generalizations of Hausdorff matrices have been made. One of them is the Endl-Jakimovski, or E-J generalization defined independently by Endl  and Jakimovski  as follows.

Let β be a real number, let $\left({\mu }_{n}\right)$ be a real sequence, and let Δ be the forward difference operator defined by $\mathrm{\Delta }{\mu }_{k}={\mu }_{k}-{\mu }_{k+1}$, ${\mathrm{\Delta }}^{n}\left({\mu }_{k}\right)=\mathrm{\Delta }\left({\mathrm{\Delta }}^{n-1}{\mu }_{k}\right)$. Then the infinite matrix $\left({H}^{\left(\beta \right)},{\mu }_{n}^{\left(\beta \right)}\right)=\left({H}^{\beta },\mu \right)=\left({h}_{nk}^{\left(\beta \right)}\right)$ is defined by

${h}_{nk}^{\left(\beta \right)}=\left\{\begin{array}{ll}\left(\begin{array}{c}n+\beta \\ n-k\end{array}\right){\mathrm{\Delta }}^{n-k}{\mu }_{k}^{\left(\beta \right)},& 0\le k\le n,\\ 0,& k>n,\end{array}$

and the associated matrix method is called a generalized Hausdorff matrix and generalized Hausdorff method, respectively. The moment sequence ${\mu }_{n}^{\left(\beta \right)}$ is given by

${\mu }_{n}^{\left(\beta \right)}={\int }_{0}^{1}{t}^{n+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(t\right),$

where $\chi \left(t\right)\in BV\left[0,1\right]$. The case $\beta =0$ corresponds to ordinary Hausdorff summability.

In a recent paper , the first author jointly with Savaş has extended the result of Das  to the E-J matrices; i.e., all conservative E-J matrices are absolutely k th power conservative for $k\ge 1$. Thereafter, Savaş and Rhoades  proved the result of Das  for double Hausdorff summability. In this paper we will consider double E-J generalization and we will prove the corresponding result of  for double E-J generalized Hausdorff matrices.

Let ${\sum }_{m=0}^{\mathrm{\infty }}{\sum }_{n=0}^{\mathrm{\infty }}{a}_{mn}$ be an infinite double series with real or complex numbers, with partial sums

${s}_{mn}=\sum _{i=0}^{m}\sum _{j=0}^{n}{a}_{ij}.$

For any double sequence $\left({u}_{mn}\right)$ we shall define

${\mathrm{\Delta }}_{11}{u}_{mn}={u}_{mn}-{u}_{m+1,n}-{u}_{m,n+1}+{u}_{m+1,n+1}.$

Denote by ${\mathcal{A}}_{k}^{2}$ the sequence space defined by

${\mathcal{A}}_{k}^{2}=\left\{{\left({s}_{mn}\right)}_{m,n=0}^{\mathrm{\infty }}:\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{\left(mn\right)}^{k-1}|{a}_{mn}{|}^{k}<\mathrm{\infty };{a}_{mn}={\mathrm{\Delta }}_{11}{s}_{m-1,n-1}\right\}$

for $k\ge 1$.

A four-dimensional matrix $T=\left({t}_{mnij}:m,n,i,j=0,1,\dots \right)$ is said to be absolutely k th power conservative for $k\ge 1$, if $T\in B\left({\mathcal{A}}_{k}^{2}\right)$; i.e., if

$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{\left(mn\right)}^{k-1}|{\mathrm{\Delta }}_{11}{s}_{m-1,n-1}{|}^{k}<\mathrm{\infty },$

then

$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{\left(mn\right)}^{k-1}|{\mathrm{\Delta }}_{11}{t}_{m-1,n-1}{|}^{k}<\mathrm{\infty },$

where

${t}_{mn}=\sum _{i=0}^{\mathrm{\infty }}\sum _{j=0}^{\mathrm{\infty }}{t}_{mnij}{s}_{ij}\phantom{\rule{1em}{0ex}}\left(m,n=0,1,\dots \right),$

see e.g. [9, 10] and the references contained therein.

A double Hausdorff matrix has entries

${h}_{mnij}=\left(\genfrac{}{}{0}{}{m}{i}\right)\left(\genfrac{}{}{0}{}{n}{j}\right){\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij},$

where $\left\{{\mu }_{ij}\right\}$ is any real or complex sequence and

${\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}=\sum _{s=0}^{m-i}\sum _{t=0}^{n-j}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m-i}{s}\right)\left(\genfrac{}{}{0}{}{n-j}{t}\right){\mu }_{i+s\cdot j+t}.$

For double Hausdorff matrices, the necessary and sufficient condition for H to be conservative is the existence of a function $\chi \left(s,t\right)\in BV\left[0,1\right]×\left[0,1\right]$ such that

${\int }_{0}^{1}{\int }_{0}^{1}|\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)|<\mathrm{\infty },$

and

${\mu }_{mn}={\int }_{0}^{1}{\int }_{0}^{1}{s}^{m}{t}^{n}\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right).$

Quite recently, Savaş and Rhoades  extended the result of Das  to double Hausdorff summability. Their theorem is as follows.

Theorem 1 

Let H be a conservative double Hausdorff matrix. Then $H\in B\left({\mathcal{A}}_{k}^{2}\right)$.

Our purpose is to achieve the result established in  for double E-J Hausdorff summability.

## Main results

The matrix ${\delta }^{\left(\alpha ,\beta \right)}=\left({\delta }_{mnij}^{\left(\alpha ,\beta \right)}\right)$, whose elements are defined by

${\delta }_{mnij}^{\left(\alpha ,\beta \right)}=\left\{\begin{array}{ll}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right),& i\le m,j\le n,\\ 0,& \text{otherwise},\end{array}$

is called a difference matrix, where α and β are real numbers.

Theorem 2 The difference matrix ${\delta }^{\left(\alpha ,\beta \right)}=\left({\delta }_{mnij}^{\left(\alpha ,\beta \right)}\right)$ is its own inverse.

Proof Let

${a}_{mnkl}=\sum _{i=0}^{m}\sum _{j=0}^{n}{\delta }_{mnij}^{\left(\alpha ,\beta \right)}{\delta }_{ijkl}^{\left(\alpha ,\beta \right)},$

thus $A={\delta }^{\left(\alpha ,\beta \right)}{\delta }^{\left(\alpha ,\beta \right)}$. For any double sequence $\left({u}_{rs}\right)$

$\begin{array}{r}\sum _{r=0}^{m}\sum _{s=0}^{n}{a}_{mnrs}{u}_{rs}\\ \phantom{\rule{1em}{0ex}}=\sum _{r=0}^{m}\sum _{s=0}^{n}\sum _{i=0}^{m}\sum _{j=0}^{n}{\delta }_{mnij}^{\left(\alpha ,\beta \right)}{\delta }_{ijrs}^{\left(\alpha ,\beta \right)}{u}_{rs}\\ \phantom{\rule{1em}{0ex}}=\sum _{r=0}^{m}\sum _{s=0}^{n}{\left(-1\right)}^{r+s}{u}_{rs}\sum _{i=r}^{m}\sum _{j=s}^{n}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\left(\genfrac{}{}{0}{}{i+\alpha }{i-r}\right)\left(\genfrac{}{}{0}{}{j+\beta }{j-s}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{r=0}^{m}\sum _{s=0}^{n}{\left(-1\right)}^{r+s}{u}_{rs}\left(\genfrac{}{}{0}{}{m+\alpha }{m-r}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-s}\right)\sum _{i=r}^{m}\sum _{j=s}^{n}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m-r}{m-i}\right)\left(\genfrac{}{}{0}{}{n-s}{n-j}\right)\\ \phantom{\rule{1em}{0ex}}={u}_{rs},\end{array}$

since

$\sum _{i=r}^{m}\sum _{j=s}^{n}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m-r}{m-i}\right)\left(\genfrac{}{}{0}{}{n-s}{n-j}\right)=\left\{\begin{array}{ll}{\left(-1\right)}^{r+s},& m=r,n=s,\\ 0,& \text{otherwise}.\end{array}$

□

Let $\left({\mu }_{mn}^{\left(\alpha ,\beta \right)}\right)$ be a given sequence and ${\mu }^{\left(\alpha ,\beta \right)}=\left({\mu }_{mnij}^{\left(\alpha ,\beta \right)}\right)$ be a diagonal matrix whose only non-zero entries are ${\mu }_{mn}^{\left(\alpha ,\beta \right)}={\mu }_{mnmn}^{\left(\alpha ,\beta \right)}$. The transformation matrix

${H}^{\left(\alpha ,\beta \right)}={\delta }^{\left(\alpha ,\beta \right)}{\mu }^{\left(\alpha ,\beta \right)}{\delta }^{\left(\alpha ,\beta \right)}$

is called a double E-J generalized Hausdorff matrix corresponding to the sequence $\left({\mu }_{mn}^{\left(\alpha ,\beta \right)}\right)$.

Theorem 3 A matrix ${H}^{\left(\alpha ,\beta \right)}=\left({h}_{mnij}^{\left(\alpha ,\beta \right)}\right)$ is a double E-J generalized Hausdorff matrix corresponding to the sequence $\left({\mu }_{mn}^{\left(\alpha ,\beta \right)}\right)$ if and only if its elements have the form

${h}_{mnij}^{\left(\alpha ,\beta \right)}=\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)},$

where

${\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)}:=\sum _{r=0}^{m-i}\sum _{s=0}^{n-j}{\left(-1\right)}^{r+s}\left(\genfrac{}{}{0}{}{m-i}{r}\right)\left(\genfrac{}{}{0}{}{n-j}{s}\right){\mu }_{i+r,j+s}^{\left(\alpha ,\beta \right)}.$

Proof Let ${H}^{\left(\alpha ,\beta \right)}={\delta }^{\left(\alpha ,\beta \right)}{\mu }^{\left(\alpha ,\beta \right)}{\delta }^{\left(\alpha ,\beta \right)}$ be a double E-J Hausdorff matrix. Applying this to a double sequence $\left({s}_{mn}\right)$ we have

$\begin{array}{rcl}{t}_{mn}& =& \sum _{i=0}^{m}\sum _{j=0}^{n}{h}_{mnij}^{\left(\alpha ,\beta \right)}{s}_{ij}\\ =& \sum _{i=0}^{m}\sum _{j=0}^{n}\sum _{r=0}^{m}\sum _{s=0}^{n}{\delta }_{mnrs}^{\left(\alpha ,\beta \right)}{\mu }_{rs}^{\left(\alpha ,\beta \right)}{\delta }_{rsij}^{\left(\alpha ,\beta \right)}{s}_{ij}\\ =& \sum _{i=0}^{m}\sum _{j=0}^{n}\sum _{r=0}^{m}\sum _{s=0}^{n}{\left(-1\right)}^{r+s}\left(\genfrac{}{}{0}{}{m+\alpha }{m-r}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-s}\right){\mu }_{rs}^{\left(\alpha ,\beta \right)}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{r+\alpha }{r-i}\right)\left(\genfrac{}{}{0}{}{s+\beta }{s-j}\right){s}_{ij}\\ =& \sum _{i=0}^{m}\sum _{j=0}^{n}{\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\sum _{r=i}^{m}\sum _{s=j}^{n}{\left(-1\right)}^{r+s}\left(\genfrac{}{}{0}{}{m-i}{m-r}\right)\left(\genfrac{}{}{0}{}{n-j}{n-s}\right){\mu }_{rs}^{\left(\alpha ,\beta \right)}{s}_{ij}\\ =& \sum _{i=0}^{m}\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\sum _{r=0}^{m-i}\sum _{s=0}^{n-j}{\left(-1\right)}^{r+s}\left(\genfrac{}{}{0}{}{m-i}{r}\right)\left(\genfrac{}{}{0}{}{n-j}{s}\right){\mu }_{i+r,j+s}^{\left(\alpha ,\beta \right)}{s}_{ij}.\end{array}$

Hence

${h}_{mnij}^{\left(\alpha ,\beta \right)}=\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\sum _{r=0}^{m-i}\sum _{s=0}^{n-j}{\left(-1\right)}^{r+s}\left(\genfrac{}{}{0}{}{m-i}{r}\right)\left(\genfrac{}{}{0}{}{n-j}{s}\right){\mu }_{i+r,j+s}^{\left(\alpha ,\beta \right)}.$

□

For double E-J Hausdorff matrices, the necessary and sufficient condition for ${H}^{\left(\alpha ,\beta \right)}$ to be conservative is the existence of a function $\chi \left(s,t\right)\in BV\left[0,1\right]×\left[0,1\right]$ such that

${\int }_{0}^{1}{\int }_{0}^{1}|\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)|<\mathrm{\infty },$

and

${\mu }_{mn}^{\left(\alpha ,\beta \right)}={\int }_{0}^{1}{\int }_{0}^{1}{s}^{m+\alpha }{t}^{n+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right).$

Theorem 4 Given a function $\chi \left(s,t\right)\in BV\left[0,1\right]×\left[0,1\right]$, a bounded variation in the unit square, the corresponding double E-J Hausdorff transformation $\left({t}_{mn}\right)$, of a sequence $\left({s}_{mn}\right)$, may be defined by

${t}_{mn}=\sum _{i=0}^{m}\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){s}_{ij}{\int }_{0}^{1}{\int }_{0}^{1}{s}^{i+\alpha }{\left(1-s\right)}^{m-i}{t}^{j+\beta }{\left(1-t\right)}^{n-j}\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right).$

Proof For $i\le m$ and $j\le n$,

$\begin{array}{rcl}{h}_{mnij}^{\left(\alpha ,\beta \right)}& =& \sum _{k=i}^{m}\sum _{l=j}^{n}{\delta }_{mnkl}^{\left(\alpha ,\beta \right)}{\mu }_{kl}^{\left(\alpha ,\beta \right)}{\delta }_{klij}^{\left(\alpha ,\beta \right)}\\ =& \sum _{k=i}^{m}\sum _{l=j}^{n}{\delta }_{mnkl}^{\left(\alpha ,\beta \right)}{\int }_{0}^{1}{\int }_{0}^{1}{s}^{k+\alpha }{t}^{l+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\cdot {\delta }_{klij}^{\left(\alpha ,\beta \right)}\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\sum _{k=i}^{m}\sum _{l=j}^{n}{\left(-1\right)}^{k+l}\left(\genfrac{}{}{0}{}{m+\alpha }{m-k}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-l}\right){\left(-1\right)}^{i+j}\left(\genfrac{}{}{0}{}{k+\alpha }{k-i}\right)\left(\genfrac{}{}{0}{}{l+\beta }{l-j}\right){s}^{k+\alpha }{t}^{l+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\sum _{k=i}^{m}\sum _{l=j}^{n}{\left(-1\right)}^{k+l+i+j}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\left(\genfrac{}{}{0}{}{m-i}{m-k}\right)\left(\genfrac{}{}{0}{}{n-j}{n-l}\right){s}^{k+\alpha }{t}^{l+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\sum _{k=0}^{m-i}\sum _{l=0}^{n-j}{\left(-1\right)}^{k+l}\left(\genfrac{}{}{0}{}{m-i}{k}\right)\left(\genfrac{}{}{0}{}{n-j}{l}\right){s}^{k+i+\alpha }{t}^{l+j+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){s}^{i+\alpha }{t}^{j+\beta }\left(\sum _{k=0}^{m-i}\sum _{l=0}^{n-j}{\left(-1\right)}^{k+l}\left(\genfrac{}{}{0}{}{m-i}{k}\right)\left(\genfrac{}{}{0}{}{n-j}{l}\right){s}^{k}{t}^{l}\right)\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){s}^{i+\alpha }{t}^{j+\beta }{\left(1-s\right)}^{m-i}{\left(1-t\right)}^{n-j}\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right).\end{array}$

□

Theorem 5 Let ${H}^{\left(\alpha ,\beta \right)}$ be a conservative double E-J Hausdorff matrix. Then ${H}^{\left(\alpha ,\beta \right)}\in B\left({\mathcal{A}}_{k}^{2}\right)$, $\alpha ,\beta \ge 0$.

As tools to prove our result, we need to the following lemmas.

Lemma 1 

Let $k\ge 1$, $n\ge v$ and $\alpha \ge 0$. Then

${E}_{m+\alpha }^{k-1}{E}_{m-\mu }^{\mu +\alpha -1}\le {E}_{\mu +\alpha }^{k-1}{E}_{m-\mu }^{\mu +\alpha +k-2}.$

The following lemma is a double version of .

Lemma 2 For $0\le s\le 1$, $0\le t\le 1$, $\alpha \ge 0$ and $\beta \ge 0$

$\sum _{i=0}^{m}\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{m+\alpha }{i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{j}\right){\left(1-s\right)}^{m}{\left(1-t\right)}^{n}{s}^{m+\alpha -i}{t}^{n+\beta -j}\le 1.$

Proof of Theorem 5 Let $\left({t}_{mn}\right)$ be the double E-J transform of a double sequence $\left({s}_{mn}\right)$; i.e.,

${t}_{mn}=\sum _{\mu =0}^{m}\sum _{v=0}^{n}{h}_{mn\mu v}^{\left(\alpha ,\beta \right)}{s}_{\mu v}.$

We will demonstrate that

$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{\left(mn\right)}^{k-1}|{a}_{mn}{|}^{k}<\mathrm{\infty }\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{\left(mn\right)}^{k-1}|{\mathrm{\Delta }}_{11}{t}_{m-1,n-1}{|}^{k}<\mathrm{\infty }.$
(1)

Write

${t}_{mn}=\sum _{\mu =0}^{m}\sum _{v=0}^{n}{b}_{\mu v}.$

Then ${b}_{mn}={\mathrm{\Delta }}_{11}{t}_{m-1,n-1}$. For $k\ge 1$

${E}_{m}^{k-1}=\left(\genfrac{}{}{0}{}{m+k-1}{m}\right)=\left(\genfrac{}{}{0}{}{m+k-1}{k-1}\right)=\frac{\left(m+k-1\right)!}{m!\left(k-1\right)!}=\frac{\mathrm{\Gamma }\left(m+k\right)}{\mathrm{\Gamma }\left(m+1\right)\mathrm{\Gamma }\left(k\right)}.$

Then

$\begin{array}{c}{E}_{m}^{k-1}\approx \frac{{m}^{k-1}}{\mathrm{\Gamma }\left(k\right)}\approx {m}^{k-1},\hfill \\ {m}^{k-1}\approx {E}_{m}^{k-1}\approx {E}_{m+\alpha }^{k-1}.\hfill \end{array}$

Due to this (1) is equivalent to

$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{a}_{mn}{|}^{k}<\mathrm{\infty }\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{b}_{mn}{|}^{k}<\mathrm{\infty }.$
(2)

For $s\in \left[0,1\right]$ and $t\in \left[0,1\right]$ define

${\varphi }_{mn}\left(s,t\right)=\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}{a}_{\mu v}.$
(3)

It follows from the Hölder inequality that

$\begin{array}{rl}|{\varphi }_{mn}\left(s,t\right){|}^{k}=& |\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}{a}_{\mu v}{|}^{k}\\ \le & \sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}|{a}_{\mu v}{|}^{k}\\ ×{\left\{\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}\right\}}^{k-1}.\end{array}$

From Lemma 2

$\begin{array}{r}\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}\\ \phantom{\rule{1em}{0ex}}=\sum _{\mu =1}^{m}\sum _{v=1}^{n}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m-\mu }\right)\left(\genfrac{}{}{0}{}{n+\beta -1}{n-v}\right){s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}\\ \phantom{\rule{1em}{0ex}}=\sum _{\mu =0}^{m-1}\sum _{v=0}^{n-1}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m-\mu -1}\right)\left(\genfrac{}{}{0}{}{n+\beta -1}{n-v-1}\right){s}^{\mu +\alpha +1}{t}^{v+\beta +1}{\left(1-s\right)}^{m-\mu -1}{\left(1-t\right)}^{n-v-1}\\ \phantom{\rule{1em}{0ex}}=st\sum _{\mu =0}^{m-1}\sum _{v=0}^{n-1}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m-\mu -1}\right)\left(\genfrac{}{}{0}{}{n+\beta -1}{n-v-1}\right){s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu -1}{\left(1-t\right)}^{n-v-1}\\ \phantom{\rule{1em}{0ex}}=O\left(st\right).\end{array}$

Hence

$|{\varphi }_{mn}\left(s,t\right){|}^{k}=O\left(1\right){\left(st\right)}^{k-1}\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}|{a}_{\mu v}{|}^{k}$

and from Lemma 1

$\begin{array}{r}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{\varphi }_{mn}\left(s,t\right){|}^{k}\\ \phantom{\rule{1em}{0ex}}=O\left(1\right)\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}{\left(st\right)}^{k-1}\\ \phantom{\rule{2em}{0ex}}×\sum _{\mu =1}^{m}\sum _{v=1}^{n}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}|{a}_{\mu v}{|}^{k}\\ \phantom{\rule{1em}{0ex}}=O\left(1\right){\left(st\right)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|{a}_{\mu v}{|}^{k}\\ \phantom{\rule{2em}{0ex}}×\sum _{m=\mu }^{\mathrm{\infty }}\sum _{n=v}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}\\ \phantom{\rule{1em}{0ex}}=O\left(1\right){\left(st\right)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|{a}_{\mu v}{|}^{k}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}\\ \phantom{\rule{2em}{0ex}}×\sum _{m=\mu }^{\mathrm{\infty }}\sum _{n=v}^{\mathrm{\infty }}{E}_{m-\mu }^{\mu +\alpha +k-2}{E}_{n-v}^{v+\beta +k-2}{\left(1-s\right)}^{m-\mu }{\left(1-t\right)}^{n-v}\\ \phantom{\rule{1em}{0ex}}=O\left(1\right){\left(st\right)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|{a}_{\mu v}{|}^{k}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}{s}^{-\mu -\alpha -k+1}{t}^{-v-\beta -k+1}\\ \phantom{\rule{1em}{0ex}}=O\left(1\right)\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}|{a}_{\mu v}{|}^{k}.\end{array}$

From Lemma of , if $\left({t}_{mn}\right)$ and $\left({\tau }_{mn}\right)$ are the $\left(H,{\mu }_{mn}\right)$ transformation of $\left({s}_{mn}\right)$ and $\left(mn{a}_{mn}\right)$, respectively, then

${\tau }_{mn}=mn{\mathrm{\Delta }}_{11}{t}_{m-1,n-1}.$

A similar consequence can be proved for $\left({H}^{\left(\alpha ,\beta \right)},{\mu }^{\left(\alpha ,\beta \right)}\right)$, see ; i.e.,

${\tau }_{mn}=\left(m+\alpha \right)\left(n+\beta \right){\mathrm{\Delta }}_{11}{t}_{m-1,n-1}.$

Hence

$\begin{array}{rl}{b}_{mn}& =\frac{1}{\left(m+\alpha \right)\left(n+\beta \right)}{\tau }_{mn}\\ =\frac{1}{\left(m+\alpha \right)\left(n+\beta \right)}\sum _{i=0}^{m}\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)}\left(i+\alpha \right)\left(j+\beta \right){a}_{ij}\\ =\sum _{i=0}^{m}\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta -1}{n-j}\right){\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)}{a}_{ij}\\ =\sum _{i=0}^{m}\sum _{j=0}^{n}{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)}{a}_{ij}.\end{array}$

Since ${H}^{\left(\alpha ,\beta \right)}$ is conservative, ${\mu }_{n}^{\left(\alpha ,\beta \right)}$ is a moment sequence,

${\mu }_{mn}^{\left(\alpha ,\beta \right)}={\int }_{0}^{1}{\int }_{0}^{1}{s}^{m+\alpha }{t}^{n+\beta }\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right),$

and

${\mathrm{\Delta }}_{1}^{m-i}{\mathrm{\Delta }}_{2}^{n-j}{\mu }_{ij}^{\left(\alpha ,\beta \right)}={\int }_{0}^{1}{\int }_{0}^{1}{s}^{i+\alpha }{\left(1-s\right)}^{m-i}{t}^{j+\beta }{\left(1-t\right)}^{n-j}\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)$

from Theorem 4. In view of (3) we can deduce that

$\begin{array}{rl}{b}_{mn}& =\sum _{i=0}^{m}\sum _{j=0}^{n}{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{\int }_{0}^{1}{\int }_{0}^{1}{s}^{i+\alpha }{\left(1-s\right)}^{m-i}{t}^{j+\beta }{\left(1-t\right)}^{n-j}\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right){a}_{ij}\\ ={\int }_{0}^{1}{\int }_{0}^{1}\left(\sum _{i=0}^{m}\sum _{j=0}^{n}{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{s}^{i+\alpha }{t}^{j+\beta }{\left(1-s\right)}^{m-i}{\left(1-t\right)}^{n-j}{a}_{ij}\right)\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right)\\ ={\int }_{0}^{1}{\int }_{0}^{1}{\varphi }_{mn}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right).\end{array}$

Using Minkowski’s inequality we get

$\begin{array}{rl}{\left\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{b}_{mn}{|}^{k}\right\}}^{1/k}& ={\left\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{\int }_{0}^{1}{\int }_{0}^{1}{\varphi }_{mn}\left(s,t\right)\phantom{\rule{0.2em}{0ex}}d\chi \left(s,t\right){|}^{k}\right\}}^{1/k}\\ \le {\int }_{0}^{1}{\int }_{0}^{1}|d\chi \left(s,t\right)|{\left\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{\varphi }_{mn}\left(s,t\right){|}^{k}\right\}}^{1/k}\\ =O\left(1\right){\int }_{0}^{1}{\int }_{0}^{1}|d\chi \left(s,t\right)|{\left\{\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}|{a}_{\mu v}{|}^{k}\right\}}^{1/k}.\end{array}$

Therefore the proof of Theorem 5 is complete. □

Specially, if we take $\alpha =0$ and $\beta =0$ in Theorem 5, we get Theorem 1 as a corollary.

The following is an example of a double E-J Hausdorff matrix.

A doubly infinite Cesàro matrix $\left(C,\gamma ,\delta \right)$ is a doubly infinite Hausdorff matrix with entries

${h}_{mnij}=\frac{\left(\genfrac{}{}{0}{}{m+\gamma -i-1}{n-i}\right)\left(\genfrac{}{}{0}{}{n+\delta -j-1}{n-j}\right)}{\left(\genfrac{}{}{0}{}{m+\gamma }{\gamma }\right)\left(\genfrac{}{}{0}{}{n+\delta }{\delta }\right)},\phantom{\rule{1em}{0ex}}\gamma ,\delta \ge 0.$

We use the following to denote the corresponding E-J generalizations of the $\left(C,\gamma ,\delta \right)$.

$\left({C}^{\left(\alpha ,\beta \right)},\gamma ,\delta \right)$ has moment sequence

${\mu }_{mn}^{\left(\alpha ,\beta \right)}={\int }_{0}^{1}{\int }_{0}^{1}{u}^{m+\alpha }{v}^{n+\beta }\gamma \delta {\left(1-u\right)}^{\gamma -1}{\left(1-v\right)}^{\delta -1}\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}dv,$

where

$\chi \left(u,v\right)=\gamma \delta {\int }_{0}^{u}{\int }_{0}^{v}{\left(1-s\right)}^{\gamma -1}{\left(1-t\right)}^{\delta -1}\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dt.$

For $i\le m$ and $j\le n$,

$\begin{array}{rcl}{h}_{mnij}^{\left(\alpha ,\beta \right)}& =& {\int }_{0}^{1}{\int }_{0}^{1}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){u}^{i+\alpha }{v}^{j+\beta }{\left(1-u\right)}^{m-i}{\left(1-v\right)}^{n-j}\phantom{\rule{0.2em}{0ex}}d\chi \left(u,v\right)\\ =& {\int }_{0}^{1}{\int }_{0}^{1}\left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)\\ ×{u}^{i+\alpha }{v}^{j+\beta }{\left(1-u\right)}^{m-i}{\left(1-v\right)}^{n-j}\gamma \delta {\left(1-u\right)}^{\gamma -1}{\left(1-v\right)}^{\delta -1}\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}dv\\ =& \gamma \delta \left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right){\int }_{0}^{1}{\int }_{0}^{1}{u}^{i+\alpha }{\left(1-u\right)}^{m-i+\gamma -1}{v}^{j+\beta }{\left(1-v\right)}^{n-j+\delta -1}\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}dv\\ =& \gamma \delta \left(\genfrac{}{}{0}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0}{}{n+\beta }{n-j}\right)B\left(i+\alpha +1,m-i+\gamma \right)B\left(j+\beta +1,n-j+\delta \right)\\ =& \frac{\gamma \mathrm{\Gamma }\left(m+\alpha +1\right)\mathrm{\Gamma }\left(m-i+\gamma \right)}{\mathrm{\Gamma }\left(m-i+1\right)\mathrm{\Gamma }\left(m+\alpha +\gamma +1\right)}\frac{\delta \mathrm{\Gamma }\left(n+\beta +1\right)\mathrm{\Gamma }\left(n-j+\delta \right)}{\mathrm{\Gamma }\left(n-j+1\right)\mathrm{\Gamma }\left(n+\beta +\delta +1\right)}\\ =& \frac{{E}_{m-i}^{\gamma -1}{E}_{n-j}^{\delta -1}}{{E}_{m+\alpha }^{\gamma }{E}_{n+\beta }^{\delta }}.\end{array}$

For the special case $\gamma ,\delta =1$,

is a double E-J Hausdorff matrix.

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## Acknowledgements

This work is supported by Istanbul Commerce University Scientific Research Projects Coordination Unit.

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Correspondence to Hamdullah Şevli.

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