# How to solve recurrence relation

How do I solve the recurrence relation in terms of $f_0$? $$f_{n+k} = -\frac{f_n}{(n+a+k)(n+b+k)}$$ where $a$ and $k$ are fixed. No idea what to do in this case due to the fact that the difference is bigger than 1 in the $f_i$. Thanks.

-
Explain to us what you mean by solving the recurrence. You want to express $f_o$ on a system of equations (possibly nonlinear) in terms of infinite variables? –  MathOverview Nov 19 '12 at 18:34

It is immediate: put $n=0$.${}{}{}{}{}$
So there are $k$ undetermined values $f_0$ up to $f_k?$ –  soup Nov 19 '12 at 21:35
Put $n=0$. We get $f_k=\frac{f(0)}{(k+a)(k+b)}$. If we prefer the subscript $n$, we get $f_n=\frac{f(0)}{(n+a)(n+b)}$. This is our general formula for $f_n$. Or is $k$ also fixed? –  André Nicolas Nov 19 '12 at 21:40
Yeah my guess is $k$ is fixed... –  soup Nov 19 '12 at 21:58
Sorry, $k$ is fixed. –  AC21 Nov 20 '12 at 9:53
Let $n=ku$. Now the equation becomes: $$f_{k(u+1)}=\frac {-f_{ku}}{(ku+a+k)(ku+b+k)}$$
So $f_n = \frac{(-1)^nf_0}{(a+k)...(a+k+n-1)(b+k)...(b+k+n-1)}$? –  soup Nov 19 '12 at 21:36
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9499521851539612, "perplexity": 484.87375396693704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042981921.1/warc/CC-MAIN-20150728002301-00096-ip-10-236-191-2.ec2.internal.warc.gz"}