Bibliography and references about approximations or definitions by recursion

Question

Im curious about the topic of approximate or write some function as a recursion, i.e., the opposite to pass a recursion to a closed form or similar things.

Im interested in these kind of topics related and different strategies involving:

• Writing a function as some limit for some recursion
• Writing a function over natural numbers as a recursion (the reverse to go from recursion to closed form, when this exists)
• Other approximations using recursions (writing summation, sequences and series as some recursion or approximation through some recursion)

Maybe this is a kind of more specific topic about numerical methods just focused in recursions.

So, can you help with some recommended bibliography (if exists)? Im an amateur but any level would be fine (I can, in any case, try to go deeper or try to learn what I need to understand some text). Thank you in advance.

Answer 1

There are many examples of functions which are defined through various methods (differential equations, integral equations, weight functions etc) and can be equivalently described through recurrence relations:

For example let us consider the case of orthogonal polynomials: If $w(x)$ is a positive function on a finite or infinite interval $(a,b)$ such that it is continuous, except possibly at a finite set of points. Let us also assume that the "moments" $\int_a^bx^nw(x)dx$ exist and are finite for any $n$. Then, a sequence $\{P_n(x)\}_{n=0}^\infty$ of polynomials $P_n(x)$, of degree $n$, such that $$ \int_a^b P_n(x)P_m(x)w(x)=\lambda_n\delta_{mn} $$ is said to be orthogonal over $(a,b)$, with respect to the weight function $w(x)$. Given this the following can be proved:

Theorem: Any sequence $\{P_n(x)\}_{n=0}^\infty$ of monic orthogonal polynomials $P_n(x)=\sum_{k=0}^na_kx^k$ is the solution of a second-order difference equation of the form $$ P_{n+1}(x)-(A_nx+B_n)P_n(x)+B_nP_{n-1}(x)=0 $$ where $A_n, B_n, C_n$ are suitable (rational) functions of the coefficients.

The above is a general result for all classes of orthogonal polynomials (Chebysev, Jacobi, Legendre, Laguerre, Hermite polynomials etc) but similar behaviours and results are typical of almost all special functions like for example hypergeometric, confluent hypergeometric, Legendre, Bessel functions etc.

You can find more details in books such as:

• Saber Elaydi, An Introduction to Difference Equations, Springer
• Nico M. Temme, Special Functions: An introduction to the classical functions of Mathematical Physics, John Willey & Sons, Inc.

and more generally in the bibliography of special functions and Mathematical Physics (Abramowitz and Stegun also contain a wealth of info).