I am looking for recursive formula for special kind of operations on power series which is called the substitution of one series into another. The main idea is the following. We have two power series:
\sum_{k=1}^\infty{b_k y^k}=\sum_{k=1}^\infty{c_k x^k}
and
y=\sum_{k=1}^\infty{a_k x^k}
I would like to obtain general formula for c_k using known coefficients a_k and b_k.
In the handbook I.S. Gradshtejn, I.M.Ryzhik "Tables of integrals, series and products", AcadPress, 2007, page 17, the formula (0.315) gives only the first four coefficients of the power series which is the substitution of one series into another.
I would be very grateful for any tips or links to papers which can give the answer for my question.
dear dr jedynak
please check the attachment if it is helpful for you.
thank you
http://mathoverflow.net/questions/53384/power-series-of-the-reciprocal-does-a-recursive-formula-exist-for-the-coeffic
Dear Mohan,
thank you for materials. I did not find the formula but the section of book and the post are very interesting and could be helpful for me during solution other problems.
Best regards
Radoslaw Jedynak
This general formula, what you are looking for, is probably Faà di Bruno's formula, and coefficients c_k are given by the combination of so-called Bell polynomials.
For the details please see that great paper about the history of the formula
http://www.maa.org/sites/default/files/images/upload_library/22/Ford/Johnson217-234.pdf
Best Regards,
Gregory
Dear Gregory,
Thank you for your clues. I have to admit that I have already read a little about Faà di Bruno's formula and it is really connected with my problem. On the other side I hope that my expected recurrence formula can be written in more clear (compact) form than proposed by mentioned author (similar to other formulas in Ryzhik).
Best regards,
Radoslaw
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