Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense
In this paper, we establish some mathematical rules for determining the initial and terminal numbers of non-zero terms in any arbitrary polynomial. These rules lead to the definitions of index $s$ and reverse index $\hat{s}$ of a polynomial. Further, building on these concepts, we introduce the orde...
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Ivan Franko National University of Lviv
2025-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/496 |
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| author | J. Banerjee A. Banerjee |
| author_facet | J. Banerjee A. Banerjee |
| author_sort | J. Banerjee |
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| description | In this paper, we establish some mathematical rules for determining the initial and terminal numbers of non-zero terms in any arbitrary polynomial. These rules lead to the definitions of index $s$ and reverse index $\hat{s}$ of a polynomial. Further, building on these concepts, we introduce the order of a polynomial $P(z)$ as $(s,\hat{s})$. If $P_{*}(z)$ is another polynomial of order $(\hat{s},s)$, then the pair $P(z)$ and $P_{*}(z)$ are referred to as symmetric polynomials. The concept of symmetric polynomials is central in this work, as we investigate the effects of weighted sharing in the wider sense (see Adv. Stud: Euro-Tbilisi Math. J., 16(4)(2023), 175-189) on the zeros of symmetric polynomials along with the sharing of poles. Our study focuses on symmetric polynomials of degree 3, analyzing their intrinsic properties. Sharing of zeros of polynomials of lower degree are critical in nature and at the same time it exhibits sophisticated structural characteristics, making them an ideal subject for such analysis. Our exploration of the sharing of zeros of symmetric polynomials establishes connections between two non-constant meromorphic functions. The article includes examples of both a general nature and specific, partial cases that serve to illustrate and validate our theoretical results. |
| format | Article |
| id | doaj-art-c2f1e12ef7ff45cb8a9ea8e041d02855 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2025-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-c2f1e12ef7ff45cb8a9ea8e041d028552025-08-20T02:40:18ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202025-03-01631486110.30970/ms.63.1.48-61496Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider senseJ. Banerjee0A. Banerjee1Department of Mathematics, University of Kalyani West Bengal, IndiaDepartment of Mathematics, University of Kalyani West Bengal, IndiaIn this paper, we establish some mathematical rules for determining the initial and terminal numbers of non-zero terms in any arbitrary polynomial. These rules lead to the definitions of index $s$ and reverse index $\hat{s}$ of a polynomial. Further, building on these concepts, we introduce the order of a polynomial $P(z)$ as $(s,\hat{s})$. If $P_{*}(z)$ is another polynomial of order $(\hat{s},s)$, then the pair $P(z)$ and $P_{*}(z)$ are referred to as symmetric polynomials. The concept of symmetric polynomials is central in this work, as we investigate the effects of weighted sharing in the wider sense (see Adv. Stud: Euro-Tbilisi Math. J., 16(4)(2023), 175-189) on the zeros of symmetric polynomials along with the sharing of poles. Our study focuses on symmetric polynomials of degree 3, analyzing their intrinsic properties. Sharing of zeros of polynomials of lower degree are critical in nature and at the same time it exhibits sophisticated structural characteristics, making them an ideal subject for such analysis. Our exploration of the sharing of zeros of symmetric polynomials establishes connections between two non-constant meromorphic functions. The article includes examples of both a general nature and specific, partial cases that serve to illustrate and validate our theoretical results.http://matstud.org.ua/ojs/index.php/matstud/article/view/496meromorphic functionsgenerating polynomialweighted sharingwider senseshared sets |
| spellingShingle | J. Banerjee A. Banerjee Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense Математичні Студії meromorphic functions generating polynomial weighted sharing wider sense shared sets |
| title | Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| title_full | Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| title_fullStr | Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| title_full_unstemmed | Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| title_short | Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| title_sort | meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense |
| topic | meromorphic functions generating polynomial weighted sharing wider sense shared sets |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/496 |
| work_keys_str_mv | AT jbanerjee meromorphicfunctionssharingthezerosoflowerdegreesymmetricpolynomialsinweightedwidersense AT abanerjee meromorphicfunctionssharingthezerosoflowerdegreesymmetricpolynomialsinweightedwidersense |