- Research
- Open Access
On solving some functional equations
- Dmitry V Kruchinin1, 2Email author
https://doi.org/10.1186/s13662-014-0347-9
© Kruchinin; licensee Springer 2015
- Received: 30 October 2014
- Accepted: 25 December 2014
- Published: 30 January 2015
Abstract
Using the notion of composita and the Lagrange inversion theorem, we present techniques for solving the following functional equation \(B(x)=H(xB(x)^{m})\), where \(H(x)\), \(B(x)\) are generating functions and \(m\in \mathbb{N}\). Also we give some examples.
Keywords
- composita
- generating function
- Lagrange inversion theorem
- functional equation
MSC
- 05A15
- 65Q20
- 39B12
1 Introduction
In this paper we study the coefficients of the powers of an ordinary generating function and their properties. Using the notion of composita, we get the solution of the functional equation \(B(x)=H(xB(x)^{m})\), which is based on the Lagrange inversion equation, where \(H(x)\), \(B(x)\) are generating functions and \(m\in\mathbb{N}\).
In the papers [1–3], the author introduced the notion of composita of a given ordinary generating function \(F(x)=\sum_{n>0}f(n)x^{n}\).
Definition 1
2 Lagrange inversion equation
In the following lemma, we give the Lagrange inversion formula, which was proved by Stanley [4].
Lemma 2
(The Lagrange inversion formula)
By using the above Lemma 2, we now give the following theorem.
Theorem 3
Proof
Next we give some examples of functional equations.
Example 4
Example 5
3 The generalized Lagrange inversion equation
Next we generalize the case \(A(x)=xH(A(x))\).
Let us introduce the following definitions.
Definition 6
Definition 7
There exists the left composita for every left composita and there exists the right composita for every right composita.
Formula (6) can be generalized for the case when a generating function is the solution of a certain functional equation. Let us prove the following theorem.
Theorem 8
Proof
Table of functional equations
Equation | Function B ( x ) | Composita \(\boldsymbol{B_{x}^{\Delta}(n,k)}\) | OEIS |
---|---|---|---|
\(B(x)=1+xB^{0}(x)\) | 1 + x | \({k \choose n-k}\) | |
\(B(x)=1+xB^{1}(x)\) | \(\frac{1}{1-x}\) | \({n-1 \choose k-1}\) | A000012 |
\(B(x)=1+xB^{2}(x)\) | \(\frac{1-\sqrt{1-4x}}{2x} \) | \(\frac{k}{n}{2n-k-1 \choose n-1}\) | A000108 |
\(B(x)=1+xB^{3}(x)\) | \(\frac{k}{n}{3n-2k \choose n-k}\) | A001764 |
Declarations
Acknowledgements
The author wishes to thank the referees for their useful comments. This work was partially supported by the Ministry of Education and Science of Russia, government order No. 1220 ‘Theoretical bases of designing informational-safe systems’.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
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