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Differentiating a Linear Recursive Sequence

Dávid Papp, Kolos Csaba Ágoston

TL;DR

The paper addresses differentiating linear recurrences with differentiable coefficient functions and derives a recurrence for the derivatives without solving for roots. It presents a constructive method based on differentiating the recurrence and solving a polynomial relation to obtain a derivative recurrence with characteristic polynomial $p(t)^2/gcd(p(t),p'(t))$. It provides an explicit, root-free procedure to compute the derivative-recurrence coefficients from the original coefficients and demonstrates the approach with Chebyshev polynomials, including higher-derivative considerations where gcd can reduce the effective order. This work extends the calculus of linear recurrences to differentiable-coefficient sequences and offers practical tools for analyzing derivatives of orthogonal polynomials and related families.

Abstract

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives also satisfies a homogeneous linear recursion whose order is at most double the order of original recursion. Similarly to the well-known operations that determine the elementwise sum and product of two linear recursive sequences, the coefficient functions of our recursion for the derivatives are easily computable from the original coefficient functions and their derivatives by direct manipulation of the coefficients of the characteristic polynomial of the recursion, without determining the roots. A simple application, computing linear recursions for derivatives of orthogonal polynomials, is presented.

Differentiating a Linear Recursive Sequence

TL;DR

The paper addresses differentiating linear recurrences with differentiable coefficient functions and derives a recurrence for the derivatives without solving for roots. It presents a constructive method based on differentiating the recurrence and solving a polynomial relation to obtain a derivative recurrence with characteristic polynomial . It provides an explicit, root-free procedure to compute the derivative-recurrence coefficients from the original coefficients and demonstrates the approach with Chebyshev polynomials, including higher-derivative considerations where gcd can reduce the effective order. This work extends the calculus of linear recurrences to differentiable-coefficient sequences and offers practical tools for analyzing derivatives of orthogonal polynomials and related families.

Abstract

Consider a sequence of real-valued functions of a real variable given by a homogeneous linear recursion with differentiable coefficients. We show that if the functions in the sequence are differentiable, then the sequence of derivatives also satisfies a homogeneous linear recursion whose order is at most double the order of original recursion. Similarly to the well-known operations that determine the elementwise sum and product of two linear recursive sequences, the coefficient functions of our recursion for the derivatives are easily computable from the original coefficient functions and their derivatives by direct manipulation of the coefficients of the characteristic polynomial of the recursion, without determining the roots. A simple application, computing linear recursions for derivatives of orthogonal polynomials, is presented.
Paper Structure (2 sections, 1 theorem, 24 equations)

This paper contains 2 sections, 1 theorem, 24 equations.

Table of Contents

  1. Introduction: the "calculus" of linear recurrence relations
  2. A linear recurrence for the sequence of derivatives

Key Result

Theorem 1

Suppose that the sequence of functions $f_0, f_1, \dots$ is defined by a linear recurrence relation of order $k$ whose characteristic polynomial is $p$, and assume that both the initial values $f_0,\dots,f_{k-1}$ and the coefficients in the recursion are differentiable. Then each $f_i$ is differenti

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Remark 2
  • Example 3: Chebyshev polynomials
  • Example 4: Second derivative sequences
  • Example 5