Context-free grammars, differential operators and formal power series

Chen, William Y. C.

doi:10.1016/0304-3975(93)90307-f

Cited by 71 publications

(83 citation statements)

References 17 publications

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“…For an alphabet A, let Q((A)) be the ring of formal Laurent series formed from letters in A. A Chen's grammar (also known as context-free grammar) over A is a function G : A → Q((A)) that replaces a letter in A with an element of Q((A)) (see [9]). The formal derivative D is a linear operator defined with respect to a Chen's grammar G. In other words, D is the unique derivation satisfying D(x) = G(x) for x ∈ A.…”

Section: Introductionmentioning

confidence: 99%

γ-positivity and partial γ-positivity of descent-type polynomials

Yeh

2019

Journal of Combinatorial Theory, Series A

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Section: Introductionmentioning

confidence: 99%

γ-positivity and partial γ-positivity of descent-type polynomials

Yeh

2019

Journal of Combinatorial Theory, Series A

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“…However, he did not seem to have noticed the connection of his formula to the work of Ramanujan. On the other hand, Dumont and Ramamonjisoa [4] use the grammatical method introduced by Chen in [3] to obtain the same combinatorial interpretation. Besides the recurrence relation (1.5) for Q n k x , Shor [5] derives the following recurrence relation and asks for a combinatorial interpretation:…”

Section: Introductionmentioning

confidence: 99%

Bijections behind the Ramanujan Polynomials

Chen

Guo

2001

Advances in Applied Mathematics

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dedicated to professor dominique foata, in honor of his 65th birthdayThe Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor's. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or the Berndt-Evans-Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan.  2001 Elsevier Science

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“…These formulas have been grammatically proved by Chen [4]. Another combinatorial proof was given by Zeilberger [26].…”

Section: Let Us Denotementioning

confidence: 90%

Cycle indicators and Special Functions

Hsu

Shiue

2001

Annals of Combinatorics

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It is shown that a class of special functions (involving Sheffer-type polynomials and Gegenbauer-Humbert-type polynomials) could have cycle indicator representations and some recurrence relations. This implies the conclusion that classical special functions with simple logarithms of generating functions can be classified this way. This paper is mainly devoted to establish various relations and identities for special functions and remarkable number sequences.

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Context-free grammars, differential operators and formal power series

Cited by 71 publications

References 17 publications

γ-positivity and partial γ-positivity of descent-type polynomials

γ-positivity and partial γ-positivity of descent-type polynomials

Bijections behind the Ramanujan Polynomials

Cycle indicators and Special Functions

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