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On mixed modular equations of degree 21

124   0   0.0 ( 0 )
 Added by S Chandankumar
 Publication date 2020
  fields
and research's language is English




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In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.



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In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $varphi(q)$ and $psi(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evaluated.
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $varphi(q)$ and $psi(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
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We study the relation between Hecke groups and the modular equations in Ramanujans theories of signature 2, 3 and 4. The solution $(alpha,beta)$ to the generalized modular equation satisfies a polynomial equation $P(alpha,beta)=0$ and we determine the degree in each of $alpha$ and $beta$ of the polynomial $P(alpha,beta)$ explicitly. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(1-beta, 1-alpha)$ is also a solution to the generalized modular equation and $P(1-beta, 1-alpha)=0$.
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92 - Bin Chen , Jie Wu 2020
In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k + 1/2 and obtain an almost type bound.
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