Coefficient Identities for Powers of Taylor and Dirichlet Series

giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a