Finite transformation formulae involving the Legendre symbol (original) (raw)

On some new modular relations for a remarkable product of theta-functions

Tbilisi Mathematical Journal, 2014

In this paper, we establish some new modular equations of degree 9. We also establish several new P-Q mixed modular equations involving theta-functions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a Remarkable product of theta-functions.

Conjectures on the evaluation of certain functions with algebraic properties

Journal of Number Theory, 2015

In this article we consider functions with Moebius-periodic Taylor expansion coefficients. These functions under some conditions take algebraic values and can be evaluated in terms of theta functions and the Dedekind eta function. Special cases are the elliptic singular moduli, the Rogers-Ramanujan continued fraction, Eisenstein series and functions associated with Jacobi symbol coefficients.

Jacobi sums and new families of irreducible polynomials of Gaussian periods

Mathematics of Computation, 2001

Let m > 2 m> 2 , ζ m \zeta _m an m m -th primitive root of 1, q ≡ 1 q\equiv 1 mod 2 m 2m a prime number, s = s q s=s_{q} a primitive root modulo q q and f = f q = ( q − 1 ) / m f=f_{q}=(q-1)/m . We study the Jacobi sums J a , b = − ∑ k = 2 q − 1 ζ m a ind s ( k ) + b ind s ( 1 − k ) J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text {ind}_{s}(k)+b\, \text {ind}_{s}(1-k)} , 0 ≤ a , b ≤ m − 1 0\leq a, b\leq m-1 , where ind s ( k ) \text {ind}_{s}(k) is the least nonnegative integer such that s ind s ( k ) ≡ k s^{\, \text {ind}_{s}(k)}\equiv k mod q q . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families P q ( x ) P_{q}(x) , q ∈ P q\in \mathcal {P} , of irreducible polynomials of Gaussian periods, η i = ∑ j = 0 f − 1 ζ q s i + m j \eta _{i}=\sum _{j=0}^{f-1}\zeta _q^{s^{i+mj}} , of degree m m , where P \mathcal {P} is a suitable set of pri...

Reflections on Symmetric Polynomials and Arithmetic Functions

Rocky Mountain Journal of Mathematics, 2005

In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and power sum symmetric polynomials. We investigate properties of these families and focus on constructing their rational roots under a product induced by convolution. A direct application of the latter is to the description of the roots of certain multiplicative arithmetic functions (the core functions) under the convolution product. The Core group is the subgroup of the group of units in the ring of arithmetic functions generated by the complete arithmetic functions.

Evaluating symplectic Gauss sums and Jacobi symbols

Nagoya mathematical journal

Stark [9] has explicitly evaluated some symplectic Gauss sums when the "denominator" matrix has odd prime level. This result is useful in computing the exact tranformation formulas of multivariable theta functions (see Stark [10], Friedberg [3] and Styer [11]). It is particularly useful when considering theta functions with quadratic forms having an odd number of variables, often a troublesome case (see Eichler [2] and Andrianov-Maloletkin [1]). Stark restricted his evaluation to symplectic Gauss sums with odd prime level denominators. In this paper, we find results analogous to the classical "reduction" theorems that allow one to decompose a Gauss sum into ones with odd prime level or level 4 or 8. To compute Gauss sums with denominators of even level, we must define 4-signatures and 8-signatures of certain submatrices. These signatures involve invariants which are more subtle than the determinants used in the odd prime level case. Consideration of Gauss sums naturally suggests the concept of Jacobi symbols. We define a symplectic Jacobi symbol, verify a number of expected properties, and finally state a reciprocity law for this symbol. In the final section, we apply our results to extend a theorem of Stark [10] calculating the transformation formulas for theta functions over algebraic number fields. Stark did the case when the lower right corner of the transformation matrix is odd, but not when it is even. By introducing the concept of 4-signatures for algebraic integers, w r e are able to handle the general case. We end with a brief comment referring to Andrianov and Maloletkin's paper [1] concerning theta functions with quadratic forms. We evaluate the multiplier system when the quadratic

Some New Families of Special Polynomials and Numbers Associated with Finite Operators

Symmetry

The aim of this study was to define a new operator. This operator unify and modify many known operators, some of which were introduced by the author. Many properties of this operator are given. Using this operator, two new classes of special polynomials and numbers are defined. Many identities and relationships are derived, including these new numbers and polynomials, combinatorial sums, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Daehee numbers, and the Changhee numbers. By applying the derivative operator to these new polynomials, derivative formulas are found. Integral representations, including the Volkenborn integral, the fermionic p-adic integral, and the Riemann integral, are given for these new polynomials.

Note on a polynomial of Emma Lehmer

Mathematics of Computation, 1991

Recently, Emma Lehmer constructed a parametric family of units in real quintic fields of prime conductor p = t 4 + 5 t 3 + 15 t 2 + 25 t + 25 p = {t^4} + 5{t^3} + 15{t^2} + 25t + 25 as translates of Gaussian periods. Later, Schoof and Washington showed that these units were fundamental units. In this note, we observe that Lehmer’s family comes from the covering of modular curves X 1 ( 25 ) → X 0 ( 25 ) {X_1}(25) \to {X_0}(25) . This gives a conceptual explanation for the existence of Lehmer’s units: they are modular units (which have been studied extensively). By relating Lehmer’s construction with ours, one finds expressions for certain Gauss sums as values of modular units on X 1 ( 25 ) {X_1}(25) .