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Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 487546, 6 pages
-Analogues of Symbolic Operators
1College of Information and Mathematical Sciences, Clayton State University, Morrow, GA 30260-0285, USA
2Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702-2900, USA
Received 1 December 2012; Revised 15 May 2013; Accepted 21 June 2013
Academic Editor: Jun Ji
Copyright © 2013 Michael J. Dancs and Tian-Xiao He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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  (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 substitutionleads to the symbolic formula(2)If, the substitutionleads to(3)If, the substitutionleads to
For a positive integer, let denote the Eulerian fraction (cf. , p. 245). It is well known that where is the th Eulerian polynomial. Additionally, (, p. 24) gives the formula Substituting leads to the formal identity
We can obtain additional identities in this fashion from other expansions of . For example, if and , we have the following analogues of (3.1)–(3.4) in : Direct proofs of (14)–(17) are given in Section 5.
Proposition 2. For a given function , define . If and ,
3. Some Applications
As an application, taking , in (19) leads to which gives The rate of convergence of this series is , much faster than whose convergence rate is .
As for a second application, we may substitute in Proposition 2, obtaining the following series transformation formulas: These four identities appear to be novel and could be used to accelerate slowly convergent alternating series . We consider them as -analogues of the ordinary Euler transformations.
4. Extensions of the Main Results
All operational formulas presented in Proposition 1 can be extended and the corresponding symbolic substitution formulas established accordingly with an analogous form of (10). For example, we may consider a generating function of the form Letting gives Applying this to the well-known identity with being the th Bernoulli number, we obtain Hence, we obtain a symbolic formula Applying this to an infinitely differentiable function at yields Similarly, using the symbolic relation we obtain another operational formula from which one may construct a series transformation formula.
As an example, substituting into (43) and noting yields the series transformation formula from
5. Selected Proofs
Equation (15) may be derived as follows:
Lemma 3. Let with , and let be any real number. One has symbolic identities involving the first Gauss series as follows: and a modified -form of Gauss’s first symbolic expression (cf. Section 127 of ):
Proof. Starting from the following Newton’s formula: we multiply to the summation from the term up and obtain Repeating the operation on the series from the term up yields The above operation is repeated from up, and so on. We obtain Substituting and into the above identity, we obtain the desired result.
Equations (16) and (17) can be proved using the first Gauss symbolic expression (49) and the following -form of the Everett’s symbolic expression (cf. , Section 129), respectively. Indeed, using (49) and noting the identity one may derive (16) as follows: Equation (17) can be proved similarly using (54). However, it can also be verified by a direct symbolic computations. In fact, we have This completes the proofs of (14)–(17).
The proof of (35) is straightforward:
The authors would like to express their gratitude to the editor and anonymous referees for their very helpful comments and suggestions.
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