scholarly journals Digit Sums and Infinite Products

2020 ◽  
Author(s):  
Samin Riasat
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Rational Function ◽  
Systematic Approach ◽  
Infinite Product ◽  
Infinite Products

Consider the sequence un defined as follows: un=+1 if the sum of the base b digits of n is even, and un=−1 otherwise, where we take b=2. Recall that the Woods-Robbins infinite product involves a rational function in n and the sequence un. Although several generalizations of the Woods-Robbins product are known in the literature, no other infinite product involving a rational function in n and the sequence un was known in closed form until recently. In this chapter we introduce a systematic approach to these products, which may be generalized to other values of b. We illustrate the approach by evaluating a large class of similar infinite products.

scholarly journals RG flows of integrable σ-models and the twist function

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
François Delduc ◽  
Sylvain Lacroix ◽  
Konstantinos Sfetsos ◽  
Konstantinos Siampos
Keyword(s):  
Renormalization Group ◽  
Large Class ◽  
Rational Function ◽  
Integrable Model ◽  
Renormalization Group Flow ◽  
Flow Equations ◽  
Non Linear ◽  
Group Flow

Abstract In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

VANISHING COEFFICIENTS IN QUOTIENTS OF THETA FUNCTIONS OF MODULUS FIVE

2020 ◽  
Vol 102 (3) ◽  
pp. 387-398
Author(s):  
SHANE CHERN ◽  
DAZHAO TANG
Keyword(s):  
Infinite Product ◽  
Theta Functions ◽  
Infinite Products ◽  
General Family

Following recent investigations of vanishing coefficients in infinite products, we show that such instances are very rare when the infinite product is among a family of theta-quotients of modulus five. We also prove that a general family of products of theta functions of modulus five can always be effectively 5-dissected.

Radial and circular slit maps of unbounded multiply connected circle domains

2008 ◽  
Vol 464 (2095) ◽  
pp. 1719-1737 ◽  
Author(s):  
T.K DeLillo ◽  
T.A Driscoll ◽  
A.R Elcrat ◽  
J.A Pfaltzgraff
Keyword(s):  
Analytic Continuation ◽  
Recent Progress ◽  
Infinite Product ◽  
Reflection Principle ◽  
Numerical Implementation ◽  
Infinite Products ◽  
Multiply Connected ◽  
Slit Domain ◽  
Circular Slit ◽  
Product Mapping

Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented.

scholarly journals Rational Divide-and-Conquer Relations

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Charinthip Hengkrawit ◽  
Vichian Laohakosol ◽  
Watcharapon Pimsert
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Rational Function ◽  
Natural Generalization ◽  
Divide And Conquer ◽  
Recursive Equation ◽  
Closed Form Solutions ◽  
Tangent Function

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form f(bn)=R(f(n),f(n),…,f(b−1)n)+g(n), where b is a positive integer ≥2; R a rational function in b−1 variables and g a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

scholarly journals CONGRUENCE PROPERTIES OF BORCHERDS PRODUCT EXPONENTS

2013 ◽  
Vol 09 (06) ◽  
pp. 1563-1578
Author(s):  
KEENAN MONKS ◽  
SARAH PELUSE ◽  
LYNNELLE YE
Keyword(s):  
Modular Forms ◽  
Logarithmic Derivative ◽  
Infinite Product ◽  
Modular Representation ◽  
Infinite Products ◽  
Analogous Statement ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Congruence Properties ◽  
Invent Math

In his striking 1995 paper, Borcherds [Automorphic forms on Os+2,2(ℝ) and infinite products, Invent. Math.120 (1995) 161–213] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant -d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, A(n,d), are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of A(n,d) modulo a prime ℓ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

scholarly journals A Systematic Approach for Calculating the Symbol Error Rate for the Entire Range of above and below the Threshold Point for the CE-OFDM System

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Monika Pinchas ◽  
Yosef Pinhasi
Keyword(s):  
Closed Form ◽  
Error Rate ◽  
Entire Range ◽  
Systematic Approach ◽  
Additive White Gaussian Noise ◽  
Density Ratio ◽  
Symbol Error Rate ◽  
Constant Envelope ◽  
High Carrier ◽  
Threshold Point

Recently, the performance of the constant envelope OFDM (CE-OFDM) was analyzed in additive white Gaussian noise (AWGN) with the help of a closed-form approximated expression for the symbol error rate (SER). This expression was obtained with the assumption of having a high carrier-to-noise ratio (CNR) which, in effect, linearized the phase demodulator (the phase demodulator was implemented with an arctangent calculator) and simplified the analysis. Thus, this expression is not accurate for the lower range of CNR. As a matter of fact, it was already observed that there is a threshold point from which the simulated SER result vanishes from the theoretically obtained expression. In this paper, we present a systematic approach for calculating the SER without assuming having the high CNR case or using linearization techniques. In other words, we derive the SER for the nonlinear case. As a byproduct, we obtain a new closed-form approximated expression for the SER based on the Laplace integral method and the Edgeworth expansion. Simulation results indicate that the simulated results and those obtained from the new derived expression are very close for the entire range of bit energy-to-noise density ratio () above and below the threshold point.

scholarly journals Wiener-Hopf Factorization for Lévy Processes Having Positive Jumps with Rational Transforms

2008 ◽  
Vol 45 (01) ◽  
pp. 118-134 ◽  
Author(s):  
Alan L. Lewis ◽  
Ernesto Mordecki
Keyword(s):  
Fourier Transform ◽  
Closed Form ◽  
Rational Function ◽  
Ruin Probability ◽  
Lévy Process ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Levy Process ◽  
Type Formula ◽  
Positive Factor

We show that the positive Wiener-Hopf factor of a Lévy process with positive jumps having a rational Fourier transform is a rational function itself, expressed in terms of the parameters of the jump distribution and the roots of an associated equation. Based on this, we give the closed form of the ruin probability for a Lévy process, with completely arbitrary negatively distributed jumps, and finite intensity positive jumps with a distribution characterized by a rational Fourier transform. We also obtain results for the ladder process and its Laplace exponent. A key role is played by the analytic properties of the characteristic exponent of the process and by a Baxter-Donsker-type formula for the positive factor that we derive.

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