Rogers Fine Identity Research Articles

Bm$(1-xy,y-x)$-expansion formula and its applications

In this paper, we first show a $(1-xy,y-x)$-expansion formula for an arbitrary analytic function $F(x)$ with respect to the basis <disp-formula><tex-math><![CDATA[$$\bigg\{\prod_{k=0}^{n-1}\frac{x-b_k}{1-x_kx}\,\bigg|\, n\geq 0\bigg\},$$]]></tex-math></disp-formula> which is mainly based on the $(f,g)$-inversion formula. Subsequently, by specializing $F(x)$, $x_n$ and $b_n$, we not only find the corresponding proofs of many classical results such as the Rogers-Fine identity, Andrews' four parametric reciprocal theorem, and Ramanujan's ${~}_1\psi_1$ summation formula, but also construct a lot of $q$-series transformations and summation formulas, including a generalization of Andrews' WP Bailey lemma.

A family of WZ pairs and q-identities

We observe that $$(F(n+k+1,k)+G(n+k,k), G(n+k,k))$$ is a WZ pair provided that (F(n, k), G(n, k)) is a WZ pair. This observation enables us to construct a bilateral sequence of WZ pairs starting from a single WZ pair. As an application, we give a one-line proof of the Rogers–Fine identity. Moreover, combing this observation and the q-WZ method for infinite series, we are able to derive a series of identities from a single q-identity. We illustrate this approach by Euler’s identity and the q-Gauss sum.

New proofs of Ramanujan’s identities on false theta functions

We provide new proofs to five of Ramanujan’s intriguing identities on false theta functions without using the Rogers-Fine identity and Bailey transforms.

Combinatorial proofs of two truncated theta series theorems

Recently, G.E. Andrews and M. Merca considered specializations of the Rogers–Fine identity and obtained partition-theoretic interpretations of two truncated identities of Gauss solving a problem by V.J.W. Guo and J. Zeng. In this paper, we provide purely combinatorial proofs of these results.

Truncated theta series and a problem of Guo and Zeng

We provide partition-theoretic interpretation of two truncated identities of Gauss solving a problem by Guo and Zeng. We also reveal that these results, together with our previous truncation of Euler's pentagonal number theorem, are essentially corollaries of the Rogers–Fine identity. Finally we examine further positivity questions related to the partition function.

Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence properties.

Euler’s Pentagonal Number Theorem and the Rogers-Fine Identity

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number theorem in his notebooks around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty of clearly presenting it with the notation at the time. In this paper we show that the method of Euler’s unpublished proof can be used to give a new proof of the celebrated Rogers-Fine identity.

Partial sums of Bailey's 6ψ6-series to faster convergent series

The modified Abel lemma on summation by parts is employed to investigate the partial sum of Bailey's very well-poised -series. Several unusual transformation formulae are established, with two of them generalizing the Rogers–Fine identity and the others involving faster convergent series.

Q-extensions of Dougall’s bilateral 2H2-series

Transformations for a bilateral 2ψ2-series are investigated by means of Abel’s lemma on summation by parts. Two q-extensions of Dougall’s classical identity for bilateral 2H2-sum are established. As by-products, the Rogers–Fine identity is recovered and a new proof is presented for Bailey’s identity of bilateral well-poised 6ψ6-series.

Some Fine combinatorics

In 1988, N.J. Fine published a monograph entitled Basic Hypergeometric Series and Applications in which he proved a number of results concerning the series F(a,b;t:q). In this paper, we present a new combinatorial interpretation for the series F(a,b;t:q) and use Fine’s work as a guide for proving the Rogers–Fine identity and many of its properties in this setting.

A NEW COMBINATORIAL STUDY OF THE ROGERS–FINE IDENTITY AND A RELATED PARTIAL THETA SERIES

We provide a transparent combinatorial derivation of a variant of the Rogers–Fine identity and a new combinatorial proof of a related partial theta series.

A bijective proof of a limiting case of Watson’s 8 φ 7 transformation formula

Andrews gave a combinatorial proof of the Rogers–Fine identity. In this paper, we present a combinatorial proof of a special case of Watson’s 8 φ 7 transformation formula, which is a generalization of Andrews’ proof.

Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm

AbstractWe present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

Parameter augmentation and the q-Gosper algorithm

We develop a method for deriving new basic hypergeometric identities from old ones by parameter augmentation. The main idea is to introduce a new parameter and use the q -Gosper algorithm to find out a suitable form of the summand. By this method, we recover some classical formulas on basic hypergeometric series and find extensions of the Rogers–Fine identity and Ramanujan’s ψ 1 1 summation formula. Moreover, we derive an identity for a ψ 3 3 summation.

A Note on The Rogers-Fine Identity

In this paper, we derive an interesting identity from the Rogers-Fine identity by applying the $q$-exponential operator method.

The q-Variations of Sylvester’s Bijection Between Odd and Strict Partitions

The q-identities corresponding to Sylvester’s bijection between odd and strict partitions are investigated. In particular, we show that Sylvester’s bijection implies the Rogers-Fine identity and give a simple proof of a partition theorem of Fine, which does not follow directly from Sylvester’s bijection. Finally, the so-called (m, c)-analogues of Sylvester’s bijection are also discussed.

Combinatorial Proofs of Identities in Ramanujan?s Lost Notebook Associated with the Rogers-Fine Identity and False Theta Functions

Ramanujan’s lost notebook contains several identities arising from the Rogers-Fine identity and/or Rogers’ false theta functions. Combinatorial proofs for many of these identities are given.

An Expansion Formula for q-Series and Applications

In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers-Fine identity, the nonterminating 6φ5 summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed.