-Analogues of Symbolic Operators

Here presented are -extensions of several linear operators including a novel -analogue of the derivative operator . Some -analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these -substitution rules may be used to construct symbolic summation and series transformation formulas, including -analogues of the classical Euler transformations for accelerating the convergence of alternating series. 1. Definitions and Basic Identities Unless otherwise stated, we consider all operators to act on formal power series in the single variable , with coefficients possibly depending on . We assume that . Issues of convergence will be addressed in a later paper. We will use to denote the identity operator and define the following operators: (1) (forward multiplicative shift), (2) (forward -difference), (3) (forward logarithmic shift). The first two of these can be regarded as -analogues of the ordinary (additive) shift and forward difference operators, respectively. will play a role similar to that of the derivative . The operator inverse of (which we denote as ) clearly exists and is equal to . We define the central -difference operator by and note that . The previous -operators are linear and satisfy some familiar identities, for example, . The binomial identity can be established by induction or by considering the operator expansion of . Treating these operators formally, we need only to consider their effect on nonnegative integer powers of . ？ ,？？ , and are “diagonal” in the sense that each maps , with the function depending on the particular operator. For example, for , and . Similarly, . With this observation, it is easy to verify many additional identities. For example, consider the alternating geometric series applied to . We have In other words, this formal power series gives the operator . Stated differently, which is exactly the result we should expect. We may establish the following identities in similar fashion: In addition to these last two identities, obeys the product rule so that is a -analogue of the ordinary derivative operator . 2. Main Results We begin with some -analogues of the symbolic substitution rules in [1] (specifically, (2.4) and (2.5)). Proposition 1. Let have the formal power series expansion , with coefficients possibly dependent on . One may obtain operational formulas according to the following rules.(1)The substitution leads to the symbolic formula (2)If , the substitution leads to (3)If , the substitution leads to Note that each of the identities in (5)–(7) can be