# Search results

1. ### Derivative of best approximation

Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...
2. ### A tricky finite series!

Hi! I've encountered the series below: \sum_{l=0}^{k-1} (r+l)^j (r+l-k)^i where r, k, i, j are positive integers and i \leq j . I am interested in expressing this series as a polynomial in k - or rather - finding the coefficients of that polynomial as i,j changes. I have reasons to...
3. ### Roots of linear sum of Fibonacci polynomials

Thanks for your reply! Interesting observations. Yes, you're right! The product of the solutions will be 2 since G can be written as the characteristic polynomial of a matrix with determinant 2. Since the solutions come in complex conjugated pairs this suggests some pretty strict bounds. The...
4. ### Roots of linear sum of Fibonacci polynomials

For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...
5. ### Implicit Hyperbolic Function

Hi Mute! Thanks for your reply. The problem is actually a result of a polynomial of degree n, which has been rewritten in it's present form. The coefficients of all n+1 terms are non-zero integers dependent on a, except for the leading term. I could probably do a more thorough analysis on the...
6. ### Implicit Hyperbolic Function

This is not a bad idea. All in all I can boil things down to tanh(ny) = cosh(y) which has the expected roots (found numerically). Solving for n is straight forward but inverting seems impossible, at least in terms of standard functions. If n is a positive integer, what can be said about y...
7. ### Implicit Hyperbolic Function

Hi tiny-tim, Thanks for your reply. I did try this and it cleans things up a bit. In particular it becomes clear that a=-2 is a convenient choice since we get 0 = (4+2a)sinh(n*y)sinh(y) + 2a[sinh(n*y) - cosh(n*y)cosh(y)] after expanding cosh((n-1)y). However, it is not clear to me how to...
8. ### Implicit Hyperbolic Function

Hi all, In studying the eigenvalues of certain tri-diagonal matrices I have encountered a problem of the following form: {(1+a/x)*2x*sinh[n*arcsinh(x/2)] - 2a*cosh[(n-1)*arcsinh(x/2)]} = 0 where a and n are constants. I'm looking to find n complex roots to this problem, but isolating x...