Questions tagged [exponential-polynomials]
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14 questions with no upvoted or accepted answers
4
votes
0
answers
302
views
Identities for powers of functions based on generalization of Lagrange interpolation
Lagrange polynomial can be used to obtain an identity:
$$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$
which holds for any integer $n>0$, any real ...
- 34.3k
2
votes
0
answers
108
views
Absolute lower bound on derivative of generalized trigonometric polynomial at zeroes
By a generalized trigonometric polynomial, I shall mean a function $f:\mathbb R^+ \rightarrow \mathbb R$ given by an expression of the form
$$f(x) := \sum_{j=1}^k a_j \cos(\alpha_j x) + b_j \sin(\...
- 739
2
votes
0
answers
402
views
lower bounding the absolute value of a determinant
In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...
- 305
1
vote
0
answers
74
views
Decidability of a polynomial-exponential equation in two variables
My question is with regards to the following (algorithmic) problem:
Problem. Given $f\in \mathbb{Z}[x,y], a,b\in \mathbb{Q}, r\in \mathbb{Z}$, do there exist positive integers $m,n$ such that $f(m,n) =...
- 151
1
vote
0
answers
128
views
Self-referencing recurrence relation with exponential
I have the self-referencing recurrence relation
$$
d(0) = 0
$$
$$
d(1) = a
$$
$$
d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4}
$$
Written as a sum:
$$
d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
- 11
1
vote
0
answers
222
views
Way to express a number in its most compact sum of powers
Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
- 11
1
vote
0
answers
102
views
Real root isolation for exponential polynomials
Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$
$$
f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t}
$$
where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...
- 1,503
1
vote
0
answers
90
views
Closed form solution for DDEs?
I am solving the equation $X−A−B e^{−Xy}−C e^{−X z}=0$ where $X, A, B$ and $C$ are 2x2 matrices and $y$ and $z$ are scalars. What will be the closed form solution ...
- 111
1
vote
0
answers
197
views
exponential and anisotropic torus
Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
- 3,472
1
vote
0
answers
104
views
Estimating decay of certain trigonometric polynomials
For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
- 959
1
vote
0
answers
69
views
Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?
When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in {\rm{...
- 113
0
votes
0
answers
804
views
How to solve definite integral involving exponential function
I am trying to get a closed form for the following definite integral:
$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 \cos^2\...
- 1
0
votes
0
answers
130
views
maximum of the sum of polynomials
Hi
I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...
- 1
0
votes
0
answers
325
views
Introductory text book for Linear Recurrence Sequences
What is a good introductory text for linear recurrence sequences?
What all are the necessary prerequisite for it? (My background is in Euclidean Fourier Analysis.) After browsing through several ...