SOME APPLICATIONS OF FINE'S THEOREM

International Journal of Scientific Research and Modern Education (IJSRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.137, ISSN (Online): 2455 - 5630, Volume 10, Issue 2, July - December, 2025 102 SOME APPLICATIONS OF FINE’S THEOREM J. D. Bulnes*, J. López-Bonilla**, S. Vidal-Beltrán** & R. Sivaraman*** * Departamento de Ciencias Exatas e Tecnologia, Universidade Federal do Amapá, Rod. Juscelino Kubitschek, Jardin Marco Zero 68903-419, Macapá, AP, Brasil ** ESIME-Zacatenco, Instituto Politécnico Nacional, Edif. 4, 1er. Piso, Col. Lindavista CP 07738, CDMX, México *** Research Associate & Founder Trustee, Pie Mathematics Association, Choolaimedu, Chennai, India Cite This Article: J. D. Bulnes, J. López-Bonilla, S. Vidal-Beltrán & R. Sivaraman, “Some Applications of Fine‟s Theorem”, International Journal of Scientific Research and Modern Education, Volume 10, Issue 2, July - December, Page Number 102-103, 2025. Copy Right: © Crystal Pen Publication, 2025 (All Rights Reserved). This is an Open Access Article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. DOI: Abstract: We exhibit how the Fine‟s theorem allows determine 𝑝(5𝑛+ 4) and 𝑟2(𝑛) if we know the corresponding partitions of n. Key Words: Fine‟s Theorem, Partition Function, Sums of Two Squares. 1. Introduction: We have the Fine‟s theorem [1-3]: If 𝜓𝑗 𝑞 = 𝐶𝑗(𝑛) ∞ 𝑛=0 𝑞𝑛then 𝜓𝑗 𝑞𝑗 = ∞ 𝑗=1 𝑅(𝑛) ∞ 𝑛=0 𝑞𝑛, (1) With: 𝑅 𝑛 = 𝐶1(𝑘1)𝐶2(𝑘2)⋯𝐶𝑛(𝑘𝑛) 𝜆 ⊢ 𝑛, (2) Where 𝜆 ⊢ 𝑛 means all partitions of n, and 𝑘𝑟 is the multiplicity of r in a given partition. In Sec. 2 we apply (1) and (2) to Ramanujan‟s “most beautiful identity” [4]: 𝑝(5𝑛+ 4) ∞ 𝑛=0 𝑞𝑛= 5 (1 − 𝑞5𝑗)5 (1 − 𝑞𝑗)6 ∞ 𝑗=1 , (3) To exhibit how the partitions of n allow to obtain the value of 𝑝 5𝑛+ 4 . In Sec. 3 we use this Fine‟s theorem to study the relation: (−1)𝑛∞ 𝑛=0 𝑟2 𝑛 𝑞𝑛= (1 − 𝑞𝑗 1 + 𝑞𝑗)2 ∞ 𝑗=1 , (4) Where 𝑟2(𝑛) is the number of representations of n as a sum of two squares [5-7] 2. Fine’s Theorem Applied to Ramanujan’s Identity (3): First we consider the functions: 𝜓𝑗 𝑞 ≔ 1−𝑞5 5 , (5) And only are different to zero the following quantities: 𝐶𝑗 5𝑚 = (−1)𝑚 5𝑚 ! 5 𝑚 , 𝑚= 0, 1, …, 5, (6) Then from (1): 𝜓𝑗(𝑞𝑗) ∞ 𝑗=1 = (1 −𝑞5𝑗)5 ∞ 𝑗=1 = 𝑅(𝑛) ∞ 𝑛=0 𝑞𝑛, 𝑅 0 = 1, (7) Where 𝑅(𝑛) is given by (2) and (6). Similarly, we introduce the functions: 𝜓 𝑗 𝑞 =1 (1 − 𝑞)6= 𝐶 𝑗(𝑛) ∞ 𝑛=0 𝑞𝑛 ∴ 𝐶 𝑗 𝑟 = 𝑟+ 5 5 , 𝑟 ≥ 0, (8) Such that: 𝜓 𝑗( ∞ 𝑗=1 𝑞𝑗) = 1 (1 − 𝑞𝑗)6 ∞ 𝑗=1 = 𝑅 (𝑛) ∞ 𝑛=0 𝑞𝑛, 𝑅 0 = 1, (9) and 𝑅 (𝑛) can be constructed with (2) and (8). Hence, from (3), (7) and (9) we deduce that 𝑝(5𝑛+ 4) is the Cauchy convolution [8] of 𝑅(𝑛) with 𝑅 𝑚 , in fact: 𝑝 5𝑛+ 4 = 5 𝑅 𝑘 𝑅 (𝑛−𝑘) 𝑛 𝑘=0 , 𝑛 ≥ 0, (10) Therefore 𝑝 4 =5, 𝑝 9 =30, 𝑝 14 =135,…; for example, if we know explicitly the 7 partitions of 5, then with (10) we can obtain that 𝑝 29 =4565. 3. Sums of Two Squares: Here we use the following functions: 𝜓𝑗 𝑞 = (1− 𝑞 )2 ∴ 𝐶𝑗 0 =1, 𝐶𝑗 1 =−2, 𝐶𝑗 2 =1, 𝐶𝑗 𝑚 =0, 𝑚 ≥3, (11) Then: 𝜓𝑗 ( 𝑞𝑗 ) ∞ 𝑗 =1 = (1− 𝑞𝑗 )2 ∞ 𝑗 =1 = 𝑅 ( 𝑛 ) ∞ 𝑛 =0 𝑞𝑛 , 𝑅 0 = 𝑅 4 =− 𝑅 2 , 𝑅 3 =− 𝑅 1 =2,… (12) With the participation of (2) and (11). In analogous manner, we consider the functions: 𝜓 𝑗 𝑞 =1 (1 + 𝑞 )2 ∴ 𝐶 𝑗 𝑛 = (−1) 𝑛 𝑛 +1 , 𝑛 ≥0, (13) Thus: International Journal of Scientific Research and Modern Education (IJSRME) International Peer Reviewed - Refereed Research Journal, Website: www.crystalpen.in Impact Factor: 7.137, ISSN (Online): 2455 - 5630, Volume 10, Issue 2, July - December, 2025 103 𝜓 𝑗 ( 𝑞𝑗 ) ∞ 𝑗 =1 = 1 (1 + 𝑞𝑗 )2 ∞ 𝑗 =1 = 𝑅 ( 𝑛 ) ∞ 𝑛 =0 𝑞𝑛 , 𝑅 0 = 𝑅 2 =1, 𝑅 1 = 𝑅 3 =−2, 𝑅 4 =4,… (14) Where (2) and (13) were applied. Hence, from (4), (12) and (14): 𝑟 2 𝑛 = (−1) 𝑛 𝑅 ( 𝑘 ) 𝑛 𝑘 =0 𝑅 𝑛 − 𝑘 , 𝑛 ≥0, (15) Then 𝑟 2 0 =1, 𝑟 2 1 = 𝑟 2 2 = 𝑟 2 4 =4, 𝑟 2 3 =0,… The Fine‟s theorem permits write many arithmetic functions in terms of integer partitions. References: 1. N. J. Fine, Basic hypergeometric series and applications, Am. Math. Soc., Providence, RI, USA (1988). 2. T. Antonelli, A surprising link between integer partitions and Euler‟s number e, Am. Math. Monthly 126, No. 5 (2019) 418-429. 3. R. L. Cruz-Simbrón, On the inverse Möbius transformation and unrestricted partitions, arXiv: 2402.07952v2 [math. NT] 22 Feb. 2024. 4. Hei-Chi Chan, An invitation to q-series. From Jacobi‟s triple product identity to Ramanujan‟s „most beautiful identity‟, World Scientific, Singapore (2011). 5. E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, New York (1985). 6. C. J. Moreno, S. S. Wagstaff Jr, Sums of squares of integers, Chapman & Hall / CRC Press, Boca Raton, Fl, USA (2006). 7. G. E. Andrews, S. Kumar Jha, J. López-Bonilla, Sums of squares, triangular numbers, and divisor sums, J. of Integer Sequences 26 (2023) Article 23.2.5. 8. R. Sivaramakrishnan, Classical theory of arithmetic functions, Marcel Dekker, New York (1989).