t: We exhibit polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k squares and k triangular numbers, respectively, and also for the color partitions

Using the Exton's multiple joint moments in four dimensional spaces due to Dirichlet density and a generalization of Bosanquet and Kestelman theorem , we prove some theorems in summability of the series containing quadruple hypergeometric functions. These theorems generalize some well known generating functions and multiplication theorems involving product of hypergeometric functions of one and more variables. We discuss some other applications and establish several interesting particular cases. Finally, we obtain an approximation formula of the series involving Exton's quadruple hypergeometric function K 11 .

Qureshi [6] proved a theorem for the degree of approximation of a periodic function $\bar f$, conjugate to a $2\pi$-periodic function $f$ and belonging to the class Lip $\theta$, by almost matrix mean of its conjugate series. The above theorem was further generalized by Qureshi and Nema [8] for a function belonging to the class $W(L^p, \Psi_1(t))$ by almost matrix mean. In the present paper we have discussed degree of approximation of above class of functions by almost $(N , p, \alpha)$ method.

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