izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2024-12-20-37izvuzmath-142Research ArticleСтатьиНЕКОТОРЫЕ ОБОБЩЕНИЯ МНОГОЧЛЕНОВ ЯКОБИ И ИХ СВОЙСТВАCertain generalizations of Jacobi polynomial and their propertiesВагелаД.WaghelaD.

Дивья Вагела

Вадодара, Гуджарат, 390001

Divya Waghela

Vadodara, Gujarat, 390001

divyarwaghela@gmail.comРаоС. Б.RaoS. B.

Снехал Б. Рао

Вадодара, Гуджарат, 390001

Snehal B. Rao

Vadodara, Gujarat, 390001

snehal.b.rao-appmath@msubaroda.ac.inУниверситет Махараджи Саяджирао в БародаИндияThe Maharaja Sayajirao University of BarodaIndia2024040120250122037Copyright © Вагела Д., Рао С.Б., 20252025Вагела Д., Рао С.Б.Waghela D., Rao S.B.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/142

Многочлен Якоби $P_{n}^{\left( \alpha ,\beta \right)}(x)$ является известным ортогональным многочленом. Изучены несколько новых свойств обобщенного многочлена Якоби $P_{n,\tau}^{\left( \alpha ,\gamma,\beta \right)}(x)$ (Waghela D., Rao S.B. A Note on Sequence of Functions associated with the Generalized Jacobi polynomial, Researches Math. 31 (2), 1-18 (2023)) и его частного случая $P_{n}^{\left( \alpha ,\gamma,\beta \right)}(x)$, которые, наряду с различными представлениями указанного обобщения, включают важное свойство ортогональности, порождающую функцию и результаты, связанные с интегральным представлением и дифференцированием обобщенного многочлена Якоби; также получен ряд известных преобразований этого обобщенного многочлена.

Jacobi polynomial $P_{n}^{\left( \alpha ,\beta \right)}(x)$ is a well-known orthogonal polynomial. In the present work, several new properties of generalized Jacobi polynomial $P_{n,\tau}^{\left( \alpha ,\gamma,\beta \right)}(x)$ (Waghela D., Rao S.B. A Note on Sequence of Functions associated with the Generalized Jacobi polynomial, Researches Math. 31 (2), 1-18 (2023)) and its special case $P_{n}^{\left( \alpha ,\gamma,\beta \right)}(x)$ have been studied, which along with different representations of the said generalization includes crucial orthogonality property, generating function, results involving integral representation, differentiation of generalized Jacobi polynomial; also many well-known transformations of this generalized polynomial have been obtained.

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The authors declare that there are no conflicts of interest present.