**Title:** About Solving Some Functional Equations related to the Lagrange Inversion Theorem

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-20-00011; **Volume 3 / Issue 1 / Year 2021**, Pages 62-69

**Document Type:** Research Paper

**Author(s):** Dmitry V. Kruchinin ^{a} , Maria Y. Perminova ^{b}

^{a}Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

^{b}Tomsk State University of Control Systems and Radioelectronics, Tomsk, Russia

Received: 1 May 2020, Accepted: 9 October 2020, Available online: 7 January 2021.

**Corresponding Author:** Dmitry V. Kruchinin (Email address: kdv@fb.tusur.ru)

**Full Text:** PDF

**Abstract**

Using the notion of the composita and the Lagrange inversion theorem, we present techniques for solving the functional equations *B*(*x*)=*H*(*x**B*(*x*)^{r}) and *A*(*F*(*x*)) = *x**H*(*A*(*x*)), where *H*(*x*), *B*(*x*) and *F*(*x*) are known generating functions and *A*(*x*) is unknown, and *r* is a rational number. The first equation is a generalization of the Lagrange inversion formula for a rational power *r* in terms of compositae. For the second equation, the recurrent solution in terms of the compositae is obtained. Also we give some examples of the obtained results.

**Keywords:** Composita, Generating function, Lagrange inversion theorem, Functional equation

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