indicial equation
Let N be an integer. Let a, b, A_1,...,A_N be constant real numbers. Let g(n) be a real function of integer variable n.
g(n) satisfies the recurrence equation:
(n + 1)(n + b)g(n+1) = (n + a)*g(n) + sum(i=1 to N)(A_i * g(n-i))
with g(n)=0 for n<0 . g(0) is obtained through boundary conditions, so it can be considered a given constant.
I need to know the general functional form of the term g(n), as a function of the given constants of the problem: a,b,N,A_1,...,A_N and g(0).
Could you help me? Could I programm in SAGE the code to provide the searched general term g(n)?
Thanks for your attention.
Javier Garcia
(homework?)
First, one may look at the particular case when the $A$-constants are all $0$. And $g(0)=1$. Then $g(1) = \frac a{1\cdot b}$, then $g(2)= \frac {a(a+1)}{2!\cdot b(b+1)}$, and so on. Looks as in a general hypergeometric dream. So it is natural to associate the formal sum: $$ F(z) = \sum_{n\ge 0} g(n)z^n $$ and to rephrase the given recursion, as an equation in $F$, that sage may eventually solve. For instance, $zF'(z) = \sum_{n\ge 1} n\,g(n)\, z^n = \sum_{n\ge 0} (n+1)\,g(n+1)\, z^{n+1} $, and $(z^bF'(z))'=\dots$
The recursion needs in fact the preceding $(N+1)$ coefficients for the one to be computed. Well, defining all negative ones as zero is ok, but then the recursion holds for $n\ge 0$. For ...(more)
In case there is a special situation of interest, please give us the special constants.
(One or more cases of interest.)
(I hope $N$ is at most two, and $a,b$ are rational.)
(As a comment here, or as an addition to the initial post.)
There is no problem to compute the first say some $100$ terms in the sequence / coefficients in the series, although the denominators will quickly grow in maginitude, and the result may be of no use. But at least writing the code will take not so much time. If this is the question, then there is an answer. A 20 minutes job, maybe. For a special situation then an answer (for guessing the other coefficients) can maybe be given, or at least guessing the shape may have a starting point. (But for the general problem...)
Thank you dan_fulea. Typically N would be <20, and the rest of the constants would not be integers. It is a pity that there is not a readily available general term. I was hoping to find out some general properties about this new function: some derivatives and simple integrals. But 100 terms in the series is not so bad: it should provide a graph good enough. I am not sure if I should dare to ask you for help. In any event, thanks for your attention.
Javier Garcia