Journal of Discrete Mathematics

Volume 2013 (2013), Article ID , 6 pages

http://dx.doi.org/10.1155/2013/487546

Research Article

-Analogues of Symbolic Operators

Michael J. Dancs¹ and Tian-Xiao He²

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.

Abstract.

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 obtained from elementary Maclaurin series by applying one of these substitution rules. We now present a less trivial example.

For a positive integer, let denote the Eulerian fraction (cf. [2], p. 245). It is well known that

where is the th Eulerian polynomial. Additionally, ([3], 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 [4]:

Direct proofs of (14)–(17) are given in Section 5.

Proposition 2. For a given function , define . If and ,

Proof. Clearly, these follow by applying the operators in (14)–(17) to the function and then evaluating at .

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.

Another extension is a -analogue of the symbolic formulas presented in [5], which is actually a Newton series type extension of the symbolic expansions given in [1]. Consider

where . We have

Finally, we present an extension of (14) using Bell polynomials (see, e.g., p. 134 in [2]) as follows:

where the values of potential Bell polynomials at are defined by

For a given function , define . From (35)–(38), we obtain series transformation formulas by simply applying (35)–(37) to :

As an example, substituting into (43) and noting

yields the series transformation formula from

5. Selected Proofs

Here we present the proofs of (14)–(17) in the sense of symbolic calculus, namely, every series expansion is considered as a formal series.

For proving (14), it suffices to make use of and (12). Indeed we have

Equation (15) may be derived as follows:

To prove (16) and (17), we first establish the following lemma.

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 [6]):

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. [6], 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:

To prove (36), we use (49) as follows:

which implies (36). Equation (37) can be proved similarly using Everett’s symbolic expression (54).

For (38), we first have

Using (39), we may write the part in the parenthesis of the rightmost term as to finish.

Acknowledgments

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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