On symbolic computation of C.P. Okeke functional equations using Python programming language

This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following func...

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Published in:	Aequationes mathematicae Vol. 98; no. 2; pp. 483 - 502
Main Authors:	Okeke, Chisom Prince, Ogala, Wisdom I., Nadhomi, Timothy
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.04.2024 Springer Nature B.V
Subjects:	Analysis Combinatorics Exact solutions Functional equations Mathematics Mathematics and Statistics Numerical methods Polynomials Programming languages Python Robustness (mathematics) Polynomial functions Functional equations Secondary: 39A70 39-04 Sagemath Python Primary: 39B52
ISSN:	0001-9054, 1420-8903
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Abstract	This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F ( a i x + b i y ) = ∑ j = 1 m ( α j x + β j y ) f ( c j x + d j y ) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered.
AbstractList	This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F ( a i x + b i y ) = ∑ j = 1 m ( α j x + β j y ) f ( c j x + d j y ) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1∑i=1nγiF(aix+biy)=∑j=1m(αjx+βjy)f(cjx+djy),for all x,y∈R, γi,αj,βj∈R, and ai,bi,cj,dj∈Q, and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, $$\begin{aligned} \textstyle \sum \limits _{i=1}^n \gamma _i F(a_i x + b_i y)=\textstyle \sum \limits _{j=1}^m(\alpha _j x + \beta _j y) f(c_j x + d_j y), \end{aligned}$$ ∑ i = 1 n γ i F ( a i x + b i y ) = ∑ j = 1 m ( α j x + β j y ) f ( c j x + d j y ) , for all $$x,y\in \mathbb {R}$$ x , y ∈ R , $$\gamma _i,\alpha _j,\beta _j \in \mathbb {R},$$ γ i , α j , β j ∈ R , and $$a_i,b_i,c_j,d_j \in \mathbb {Q},$$ a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered.
Author	Ogala, Wisdom I. Okeke, Chisom Prince Nadhomi, Timothy
Author_xml	– sequence: 1 givenname: Chisom Prince orcidid: 0000-0002-9034-408X surname: Okeke fullname: Okeke, Chisom Prince email: chisom.okeke@us.edu.pl organization: Institute of Mathematics, University of Silesia – sequence: 2 givenname: Wisdom I. surname: Ogala fullname: Ogala, Wisdom I. organization: Department of Mathematics, University of Central Florida – sequence: 3 givenname: Timothy surname: Nadhomi fullname: Nadhomi, Timothy organization: Institute of Mathematics, University of Silesia
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SubjectTerms	Analysis Combinatorics Exact solutions Functional equations Mathematics Mathematics and Statistics Numerical methods Polynomials Programming languages Python Robustness (mathematics)
Title	On symbolic computation of C.P. Okeke functional equations using Python programming language
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