Functions on the Plane as Combinations of Powers of Distances to Points

and Applied Analysis 3 We use the formula of the product of two series ∑∞ n 0 an and ∑∞ n 0 bn: ∞ ∑ n 0 an ∞ ∑ n 0 bn ∞ ∑ n 0 n ∑ k 0 akbn−k. 2.3 The right side of formula 2.2 becomes ∞ ∑ i 0 i ∑ j 0 aij t i jxi−j · ∞ ∑ d 0 tpd x ∞ ∑ i 0 i ∑ k 0 i−k ∑ j 0 ai−k,jpk x t jxi−k−j . 2.4 Let us change the order of addition with respect to the indices j and k: i ∑ k 0 i−k ∑ j 0 ai−k,jpk x t jxi−k−j i ∑ j 0 i−j ∑ k 0 ai−k,jpk x t jxi−k−j . 2.5 The following formula is formula 8.911 of 1 : pk x k/2 ∑ l 0 bklx k−2l, 2.6