Some of the more prominent mathematicians to make contributions to this field include Jacobi, Perron, Hermite, Gauss, Cauchy, and Stieljes. By the beginning of the 20th century, the discipline had greatly advanced from the initial work of Wallis. Since the beginning of the 20th century continued fractions have made their appearances in other fields. This brief sketch into the past of continued fractions is intended to provide an overview of the development of this field.
Rogers-Ramanujan Continued Fraction
Though its initial development seems to have taken a long time, once started, the field and its analysis grew rapidly. Srinivasa Ramanujan, an Indian Mathematician had an extreme fondness for continued fractions. He expressed continued fractions in terms of infinite products. Andrews represented these infinite product representations of continued fractions in terms of dissections which paved the way for further developments in dissections. In the intervening years, several studies have been conducted on dissections of continued fractions and many research papers have been published.
This paper gives a brief note on the developments of dissections of continued fractions in the field of Mathematics. The 2-dissections of R q and R q -1 were given by Andrews. The following are the 2-dissections, 2. These 4-dissections were first proved by Lewis and Liu. Hirschhorn also proved the following 5-dissections of R q and R q -1 : 2. Basil Gordon showed that 2. Hirschhorn proved the following theorem; For 2. He demonstrated the periodicity of the sign of the coefficients in G q and its reciprocal and showed that certain coefficients are zero, which are already proved by Richmond and Szekeres.
The 8-dissection for G q and its reciprocal given by Hirschhorn are 2. The 2-dissections of RC q and its reciprocal are given by 2. He proved that, if then 2. Bernard studied the 2-, 3-, 4-, 6-, and dissections of a continued fraction of order 12, and proved that when the above mentioned continued fraction and its reciprocal are expanded as power series, the sign of the coefficients are periodic with period Conclusions In the foregoing sections, results concerning dissections of various continued fractions are presented.
However, there are other continued fractions for which dissections are not yet given by researchers. Many researchers have worked on dissections and studied the periodicity of the coefficient.
Rocky Mountain Journal of Mathematics
We believe that these results can further be extended for higher dissections of various continued fractions. Kislyn, Random House: New York. Blanton, Springer — Verlag, NewYork.
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Oxford University Press: New York. Springer-Verlag: New York. Mat h. London Math. The Ramanujan Journal, 4: Xia Search this author in:.
Rogers–Ramanujan continued fraction - Wikipedia
The 2- and 4-dissections of some infinite products are established in this paper. As corollaries of our results, we derive the 4-dissections of some continued fractions appearing in Ramanujan's notebooks and their reciprocals. Source Rocky Mountain J. Zentralblatt MATH identifier Keywords Ramanujan's continued fraction theta functions infinite product. Xia, Ernest X. On 2- and 4-dissections for some infinite products. Rocky Mountain J. Proofs of these identities have been given in [1, pp.
We will prove 20 and The identity 21 may be obtained in a similar way, starting by multiply- ing 22 by It would be interesting to have direct proofs of these results. Identities involving modular forms In this section we outline the basic properties of the modular forms u, v, y and z and their level 13 analogues U , V , Y and Z. All of these forms have weight 2 except for Y which has weight 6. We begin by noting some known transformation formulas involving S, U and V. For i , see [2, p. The identity ii follows immediately from the definition of S and the transformation formula for the Dedekind eta function in 9.
For iii — vi , see [11, Th. They imply infinite product formulas for u and v, respectively, and on comparing them with the infinite products that occur in the definitions of s and y, we immediately deduce the identities 24 i and ii. The identity 24 iii was known to Ramanujan; see [1, p. It can be proved by applying the d operator q dq to the identity 10 and appealing the definitions of v and z.
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This completes the proof of the identities in By Lemma 2. It follows that there are unique constants, a8 , a7 ,. Y 0 It follows that h is bounded. R S On applying 25 ii we complete the proof.
Eisenstein series In the lost notebook [27, pp. The results may be stated as: Theorem 5. Let M and N be the Eisenstein series defined in Section 2. The goal of this section is to prove the level 13 analogues given by Theorem 5. The method of proof will be to adapt the procedure developed in  where the analogous results for levels 5 and 7 were derived.
By Theo- rem 4. To simplify the calculations we will work in terms of S and U and then use Theorem 4. We will not list any level 5 results in this section; for these, as well as the level 7 results, the reader is referred to . By the definition of Z in Section 2. Next, by the chain rule and the definition of U in Section 2.
These are immediate consequences of the definitions of S and Y in Section 2.
These are immediate consequences of the identity 6 and Lemma 5. Take the logarithmic derivative of the first identity in Lemma 5. The identity 31 may be obtained similarly, starting from the second identity in Lemma 5. Finally, the identity 32 may be obtained by subtracting 31 from By 26 , the right hand side of 35 is identically zero. Hence, we de- duce The results of Lemmas 5. On using these in 40 we deduce 36 — Use 25 i to eliminate U from each of 36 — The j-invariant defined in Section 2.
S 13 Proof. Substitute the results of Theorem 5. Many transformation formulas of degree 13 for hypergeometric functions can be deduced from Theorem 5. The next result gives two of the simplest transformation formulas. Theorem 5. From [10, 2. Differential equations The goal of this section is to show that Z satisfies a third order linear differential equation with respect to T. We begin by constructing a second order nonlinear differential equation for U in terms of R.
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Theorem 6. The derivative of the second term on the right hand side may be found by direct calculation. On simplifying and dividing both sides by U we complete the proof. Apply the operator R dR to both sides of the differential equation in Theorem 6.