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 [1] in 1933. Later a few authors studied double Hausdorff matrices; see e.g. Ramanujan [2] and Ustina [3].

Several generalizations of Hausdorff matrices have been made. One of them is the Endl-Jakimovski, or E-J generalization defined independently by Endl [4] and Jakimovski [5] as follows.

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

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

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

$${\mu }_n^{(\beta )}={\int }_0^1{t}^{n+\beta }d\chi (t),$$

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

In a recent paper [6], the first author jointly with Savaş has extended the result of Das [7] 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 [8] proved the result of Das [7] for double Hausdorff summability. In this paper we will consider double E-J generalization and we will prove the corresponding result of [6] 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 $(u_{mn})$ 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

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

for $k\ge 1$.

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

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

then

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

where

$$t_{mn}=\sum _{i=0}^{\mathrm{\infty }}\sum _{j=0}^{\mathrm{\infty }}{t}_{mnij}{s}_{ij}(m,n=0,1,\dots ),$$

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

A double Hausdorff matrix has entries

$$h_{mnij}=\left(\genfrac{}{}{0ex}{}{m}{i}\right)\left(\genfrac{}{}{0ex}{}{n}{j}\right){\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij},$$

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

$${\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}=\sum _{s=0}^{m-i}\sum _{t=0}^{n-j}{(-1)}^{i+j}\left(\genfrac{}{}{0ex}{}{m-i}{s}\right)\left(\genfrac{}{}{0ex}{}{n-j}{t}\right){\mu }_{i+s\cdot j+t}.$$

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

$${\int }_0^1{\int }_0^1|d\chi (s,t)|<\mathrm{\infty },$$

and

$${\mu }_{mn}={\int }_0^1{\int }_0^1{s}^m{t}^{n}d\chi (s,t).$$

Quite recently, Savaş and Rhoades [8] extended the result of Das [7] to double Hausdorff summability. Their theorem is as follows.

Theorem 1 [8]

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

Our purpose is to achieve the result established in [7] for double E-J Hausdorff summability.

Main results

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

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

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

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

Proof Let

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

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

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

since

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

 □

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

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

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

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

$$h_{mnij}^{(\alpha ,\beta )}=\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right){\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )},$$

where

$${\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )}:=\sum _{r=0}^{m-i}\sum _{s=0}^{n-j}{(-1)}^{r+s}\left(\genfrac{}{}{0ex}{}{m-i}{r}\right)\left(\genfrac{}{}{0ex}{}{n-j}{s}\right){\mu }_{i+r,j+s}^{(\alpha ,\beta )}.$$

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

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

Hence

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

 □

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

$${\int }_0^1{\int }_0^1|d\chi (s,t)|<\mathrm{\infty },$$

and

$${\mu }_{mn}^{(\alpha ,\beta )}={\int }_0^1{\int }_0^1{s}^{m+\alpha }{t}^{n+\beta }d\chi (s,t).$$

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

$$t_{mn}=\sum _{i=0}^m\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)s_{ij}{\int }_0^1{\int }_0^1{s}^{i+\alpha }{(1-s)}^{m-i}{t}^{j+\beta }{(1-t)}^{n-j}d\chi (s,t).$$

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

$$\begin{array}{rcl}{h}_{mnij}^{(\alpha ,\beta )}& =& \sum _{k=i}^m\sum _{l=j}^n{\delta }_{mnkl}^{(\alpha ,\beta )}{\mu }_{kl}^{(\alpha ,\beta )}{\delta }_{klij}^{(\alpha ,\beta )}\\ =& \sum _{k=i}^m\sum _{l=j}^n{\delta }_{mnkl}^{(\alpha ,\beta )}{\int }_0^1{\int }_0^1{s}^{k+\alpha }{t}^{l+\beta }d\chi (s,t)\cdot {\delta }_{klij}^{(\alpha ,\beta )}\\ =& {\int }_0^1{\int }_0^1\sum _{k=i}^m\sum _{l=j}^n{(-1)}^{k+l}\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-k}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-l}\right){(-1)}^{i+j}\left(\genfrac{}{}{0ex}{}{k+\alpha }{k-i}\right)\left(\genfrac{}{}{0ex}{}{l+\beta }{l-j}\right)s^{k+\alpha }{t}^{l+\beta }d\chi (s,t)\\ =& {\int }_0^1{\int }_0^1\sum _{k=i}^m\sum _{l=j}^n{(-1)}^{k+l+i+j}\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)\left(\genfrac{}{}{0ex}{}{m-i}{m-k}\right)\left(\genfrac{}{}{0ex}{}{n-j}{n-l}\right)s^{k+\alpha }{t}^{l+\beta }d\chi (s,t)\\ =& {\int }_0^1{\int }_0^1\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)\sum _{k=0}^{m-i}\sum _{l=0}^{n-j}{(-1)}^{k+l}\left(\genfrac{}{}{0ex}{}{m-i}{k}\right)\left(\genfrac{}{}{0ex}{}{n-j}{l}\right)s^{k+i+\alpha }{t}^{l+j+\beta }d\chi (s,t)\\ =& {\int }_0^1{\int }_0^1\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)s^{i+\alpha }{t}^{j+\beta }\left(\sum _{k=0}^{m-i}\sum _{l=0}^{n-j}{(-1)}^{k+l}\left(\genfrac{}{}{0ex}{}{m-i}{k}\right)\left(\genfrac{}{}{0ex}{}{n-j}{l}\right)s^k{t}^l\right)d\chi (s,t)\\ =& {\int }_0^1{\int }_0^1\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)s^{i+\alpha }{t}^{j+\beta }{(1-s)}^{m-i}{(1-t)}^{n-j}d\chi (s,t).\end{array}$$

 □

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

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

Lemma 1 [6]

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 [11].

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

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

Proof of Theorem 5 Let $(t_{mn})$ be the double E-J transform of a double sequence $(s_{mn})$; i.e.,

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

We will demonstrate that

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

(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 $k\ge 1$

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

Then

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

Due to this (1) is equivalent to

$$\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|a_{mn}{|}^k<\mathrm{\infty }\Rightarrow \sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|b_{mn}{|}^k<\mathrm{\infty }.$$

(2)

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

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

(3)

It follows from the Hölder inequality that

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

From Lemma 2

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

Hence

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

and from Lemma 1

$$\begin{array}{r}\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|{\varphi }_{mn}(s,t){|}^k\\ =O(1)\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}{(st)}^{k-1}\\ \times \sum _{\mu =1}^m\sum _{v=1}^n{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{s}^{\mu +\alpha }{t}^{v+\beta }{(1-s)}^{m-\mu }{(1-t)}^{n-v}|a_{\mu v}{|}^k\\ =O(1){(st)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|a_{\mu v}{|}^k\\ \times \sum _{m=\mu }^{\mathrm{\infty }}\sum _{n=v}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}{E}_{m-\mu }^{\mu +\alpha -1}{E}_{n-v}^{v+\beta -1}{(1-s)}^{m-\mu }{(1-t)}^{n-v}\\ =O(1){(st)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|a_{\mu v}{|}^k{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}\\ \times \sum _{m=\mu }^{\mathrm{\infty }}\sum _{n=v}^{\mathrm{\infty }}{E}_{m-\mu }^{\mu +\alpha +k-2}{E}_{n-v}^{v+\beta +k-2}{(1-s)}^{m-\mu }{(1-t)}^{n-v}\\ =O(1){(st)}^{k-1}\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{s}^{\mu +\alpha }{t}^{v+\beta }|a_{\mu v}{|}^k{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}{s}^{-\mu -\alpha -k+1}{t}^{-v-\beta -k+1}\\ =O(1)\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}|a_{\mu v}{|}^k.\end{array}$$

From Lemma of [8], if $(t_{mn})$ and $({\tau }_{mn})$ are the $(H,{\mu }_{mn})$ transformation of $(s_{mn})$ and $(mn{a}_{mn})$, respectively, then

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

A similar consequence can be proved for $(H^{(\alpha ,\beta )},{\mu }^{(\alpha ,\beta )})$, see [6]; i.e.,

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

Hence

$$\begin{array}{rl}{b}_{mn}& =\frac{1}{(m+\alpha )(n+\beta )}{\tau }_{mn}\\ =\frac{1}{(m+\alpha )(n+\beta )}\sum _{i=0}^m\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right){\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )}(i+\alpha )(j+\beta )a_{ij}\\ =\sum _{i=0}^m\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{m+\alpha -1}{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta -1}{n-j}\right){\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )}{a}_{ij}\\ =\sum _{i=0}^m\sum _{j=0}^n{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )}{a}_{ij}.\end{array}$$

Since $H^{(\alpha ,\beta )}$ is conservative, ${\mu }_n^{(\alpha ,\beta )}$ is a moment sequence,

$${\mu }_{mn}^{(\alpha ,\beta )}={\int }_0^1{\int }_0^1{s}^{m+\alpha }{t}^{n+\beta }d\chi (s,t),$$

and

$${\mathrm{\Delta }}_1^{m-i}{\mathrm{\Delta }}_2^{n-j}{\mu }_{ij}^{(\alpha ,\beta )}={\int }_0^1{\int }_0^1{s}^{i+\alpha }{(1-s)}^{m-i}{t}^{j+\beta }{(1-t)}^{n-j}d\chi (s,t)$$

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

$$\begin{array}{rl}{b}_{mn}& =\sum _{i=0}^m\sum _{j=0}^n{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{\int }_0^1{\int }_0^1{s}^{i+\alpha }{(1-s)}^{m-i}{t}^{j+\beta }{(1-t)}^{n-j}d\chi (s,t)a_{ij}\\ ={\int }_0^1{\int }_0^1\left(\sum _{i=0}^m\sum _{j=0}^n{E}_{m-i}^{i+\alpha -1}{E}_{n-j}^{j+\beta -1}{s}^{i+\alpha }{t}^{j+\beta }{(1-s)}^{m-i}{(1-t)}^{n-j}{a}_{ij}\right)d\chi (s,t)\\ ={\int }_0^1{\int }_0^1{\varphi }_{mn}(s,t)d\chi (s,t).\end{array}$$

Using Minkowski’s inequality we get

$$\begin{array}{rl}{\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}|b_{mn}{|}^k\}}^{1/k}& ={\left\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}\right|{\int }_0^1{\int }_0^1{\varphi }_{mn}(s,t)d\chi (s,t){|}^k\}}^{1/k}\\ \le {\int }_0^1{\int }_0^1|d\chi (s,t)|{\left\{\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{E}_{m+\alpha }^{k-1}{E}_{n+\beta }^{k-1}\right|{\varphi }_{mn}(s,t){|}^k\}}^{1/k}\\ =O(1){\int }_0^1{\int }_0^1|d\chi (s,t)|{\{\sum _{\mu =1}^{\mathrm{\infty }}\sum _{v=1}^{\mathrm{\infty }}{E}_{\mu +\alpha }^{k-1}{E}_{v+\beta }^{k-1}|a_{\mu v}{|}^k\}}^{1/k}.\end{array}$$

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

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

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

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

$${\mu }_{mn}^{(\alpha ,\beta )}={\int }_0^1{\int }_0^1{u}^{m+\alpha }{v}^{n+\beta }\gamma \delta {(1-u)}^{\gamma -1}{(1-v)}^{\delta -1}dudv,$$

where

$$\chi (u,v)=\gamma \delta {\int }_0^u{\int }_0^v{(1-s)}^{\gamma -1}{(1-t)}^{\delta -1}dsdt.$$

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

$$\begin{array}{rcl}{h}_{mnij}^{(\alpha ,\beta )}& =& {\int }_0^1{\int }_0^1\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)u^{i+\alpha }{v}^{j+\beta }{(1-u)}^{m-i}{(1-v)}^{n-j}d\chi (u,v)\\ =& {\int }_0^1{\int }_0^1\left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)\\ \times u^{i+\alpha }{v}^{j+\beta }{(1-u)}^{m-i}{(1-v)}^{n-j}\gamma \delta {(1-u)}^{\gamma -1}{(1-v)}^{\delta -1}dudv\\ =& \gamma \delta \left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right){\int }_0^1{\int }_0^1{u}^{i+\alpha }{(1-u)}^{m-i+\gamma -1}{v}^{j+\beta }{(1-v)}^{n-j+\delta -1}dudv\\ =& \gamma \delta \left(\genfrac{}{}{0ex}{}{m+\alpha }{m-i}\right)\left(\genfrac{}{}{0ex}{}{n+\beta }{n-j}\right)B(i+\alpha +1,m-i+\gamma )B(j+\beta +1,n-j+\delta )\\ =& \frac{\gamma \mathrm{\Gamma }(m+\alpha +1)\mathrm{\Gamma }(m-i+\gamma )}{\mathrm{\Gamma }(m-i+1)\mathrm{\Gamma }(m+\alpha +\gamma +1)}\frac{\delta \mathrm{\Gamma }(n+\beta +1)\mathrm{\Gamma }(n-j+\delta )}{\mathrm{\Gamma }(n-j+1)\mathrm{\Gamma }(n+\beta +\delta +1)}\\ =& \frac{E_{m-i}^{\gamma -1}{E}_{n-j}^{\delta -1}}{E_{m+\alpha }^{\gamma }{E}_{n+\beta }^{\delta }}.\end{array}$$

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

$$(C^{(\alpha ,\beta )},1,1)=\{\begin{array}{ll}\frac{1}{(m+\alpha +1)(n+\beta +1)},& i\le m\text{ and }j\le n,\\ 0,& \text{otherwise}\end{array}$$

is a double E-J Hausdorff matrix.