Hecke Operators on Jacobi-like Forms

2001 ◽  
Vol 44 (3) ◽  
pp. 282-291 ◽  
Author(s):  
Min Ho Lee ◽  
Hyo Chul Myung
Keyword(s):  
Functional Equation ◽  
Power Series ◽  
Explicit Formula ◽  
Formal Power Series ◽  
Modular Forms ◽  
Discrete Subgroup ◽  
Hecke Operators ◽  
Hecke Operator ◽  
Upper Half Plane ◽  
Formal Power

AbstractJacobi-like forms for a discrete subgroup are formal power series with coefficients in the space of functions on the Poincaré upper half plane satisfying a certain functional equation, and they correspond to sequences of certain modular forms. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

scholarly journals Jacobi-like forms and power series bundles

2002 ◽  
Vol 66 (2) ◽  
pp. 301-311 ◽  
Author(s):  
Min Ho Lee
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Half Plane ◽  
Modular Forms ◽  
Vector Bundles ◽  
Discrete Subgroup ◽  
Holomorphic Functions ◽  
Jacobi Forms ◽  
Upper Half Plane ◽  
Formal Power

Jacobi-like forms are certain formal power series which generalise Jacobi forms in some sense, and they are closely linked to modular forms when their coefficients are holomorphic functions on the Poincaré upper half plane. We construct two types of vector bundles whose fibres are isomorphic to the space of certain formal power series and whose sections can be identified with Jacobi-like forms for a discrete subgroup of SL (2,ℝ).

ON OHTSUKI'S INVARIANTS OF HOMOLOGY 3-SPHERES

1999 ◽  
Vol 08 (08) ◽  
pp. 1049-1063 ◽  
Author(s):  
RUTH J. LAWRENCE
Keyword(s):  
Power Series ◽  
Explicit Formula ◽  
Formal Power Series ◽  
Original Work ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Rational Homology ◽  
Formal Power

By analysing Ohtsuki's original work in which he produced a formal power series invariant of rational homology 3-spheres, we obtain a simplified explicit formula for them, which may also be compared with Rozansky's integral expression. We further show their relation to the exact SO(3) Witten-Reshetikhin-Turaev invariants at roots of unity in a stronger form than that given in Ohtsuki's original work.

On Covariant Embeddings of a Linear Functional Equation with Respect to an Analytic Iteration Group

2003 ◽  
Vol 13 (07) ◽  
pp. 1853-1875 ◽  
Author(s):  
Harald Fripertinger ◽  
Ludwig Reich
Keyword(s):  
Functional Equation ◽  
Power Series ◽  
Boundary Conditions ◽  
Large Class ◽  
Formal Power Series ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Iteration Group ◽  
Analytic Iteration ◽  
Formal Power

Let a(x), b(x), p(x) be formal power series in the indeterminate x over [Formula: see text] (i.e. elements of the ring [Formula: see text] of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group [Formula: see text] in [Formula: see text]. By a covariant embedding of the linear functional equation [Formula: see text] (for the unknown series [Formula: see text]) with respect to [Formula: see text]. In this paper we solve the system ((Co1), (Co2)) (of so-called cocycle equations) completely, describe when and how the boundary conditions (B1) and (B2) can be satisfied, and present a large class of equations (L) together with iteration groups [Formula: see text] for which there exist covariant embeddings of (L) with respect to [Formula: see text].

scholarly journals JACOBI-LIKE FORMS, PSEUDODIFFERENTIAL OPERATORS, AND GROUP COHOMOLOGY

2008 ◽  
Vol 78 (1) ◽  
pp. 55-71
Author(s):  
MIN HO LEE
Keyword(s):  
Modular Forms ◽  
Pseudodifferential Operators ◽  
Discrete Subgroup ◽  
Group Cohomology ◽  
Derivative Operator ◽  
Formal Laurent Series ◽  
Hecke Operator ◽  
Linear Maps ◽  
Formal Inverse ◽  
Upper Half Plane

AbstractPseudodifferential operators are formal Laurent series in the formal inverse ∂−1 of the derivative operator ∂ whose coefficients are holomorphic functions on the Poincaré upper half-plane. Given a discrete subgroup Γ of SL(2,ℝ), automorphic pseudodifferential operators for Γ are pseudodifferential operators that are Γ-invariant, and they are closely linked to Jacobi-like forms and modular forms for Γ. We construct linear maps from the space of automorphic pseudodifferential operators and from the space of Jacobi-like forms for Γ to the cohomology space of the group Γ, and prove that these maps are compatible with the respective Hecke operator actions.

scholarly journals THE ACTION OF HECKE OPERATORS ON HYPERGEOMETRIC FUNCTIONS

2010 ◽  
Vol 89 (1) ◽  
pp. 51-74 ◽  
Author(s):  
VICTOR H. MOLL ◽  
SINAI ROBINS ◽  
KIRK SOODHALTER
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Hypergeometric Functions ◽  
Hecke Operators ◽  
Formal Power

AbstractWe study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na:n∈ℕ,a∈ℤ}, and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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