ФУНКЦИОНАЛЬНЫЕ СООТНОШЕНИЯ, ВЫВОДИМЫЕ ИЗ ОПРЕДЕЛЕНИЯ ГИПЕРГЕОМЕТРИЧЕСКОЙ ФУНКЦИЙ АППЕЛЯ ( )

363

В этой статье рассматривается новая взаимосвязь между гипергеометрической функцией Аппеля
(
)
1 1 2
F a;b ,b ;c; x, y и функцией Кампе де Фериет F44;3;4,3,4 [x, y] . Рассмотрено метод разложение на ряд гипергеометрической функции Аппеля, а также использовано метод группировки членов равномерно сходящегося степенного ряда. Из полученного соотношения получено, что существуют связь между функциями Гаусса, обобщенной гипергеометрической функцией 8 F7(x4) и гиперболическими косинусами.

Журнал номиFarDU Ilmiy Xabarlari
Нашр номи2021-6 Ilova
Кўришлар сони 363

Internet ҳавола

DOI

UzSCI тизимида яратилган сана 18-10-2022

Ўқишлар сони 0

Нашр санаси 11-02-2022

Мақола тилиRus

Саҳифалар сони6-14

Калит сўзлар

гипергеометрическиe функции

функцией Аппеля

Ўзбек

Мақолада Аппелнинг ( )
1 1 2
F a;b ,b ;c; x, y гипергеометрик функцияси ва Кампе де Фериетнинг F44;3;4,3,4 [x, y] функцияси орасидаги янги муносабат олинган. Бунда Аппелнинг гипергеометрик функциясининг қаторга ёйилмасидан ҳамда текис яқинлашувчи даражали қаторларнинг ҳадларини группалаш усулидан фойдаланилган. Олинган муносабатдан Гаусс функциялари, умумлашган гипергеометрик 8 F7(x4) функция ва гиперболик косинуслар ўртасидаги боғланишлар мавжудлиги аниқланган.

Калит сўзлар

гипергеометрик фунциялар

Аппел функцияси

Русский

В этой статье рассматривается новая взаимосвязь между гипергеометрической функцией Аппеля
(
)
1 1 2
F a;b ,b ;c; x, y и функцией Кампе де Фериет F44;3;4,3,4 [x, y] . Рассмотрено метод разложение на ряд гипергеометрической функции Аппеля, а также использовано метод группировки членов равномерно сходящегося степенного ряда. Из полученного соотношения получено, что существуют связь между функциями Гаусса, обобщенной гипергеометрической функцией 8 F7(x4) и гиперболическими косинусами.

Калит сўзлар

гипергеометрическиe функции

функцией Аппеля

English

In this paper, we get a new relationship between Appel’s hypergeometric function F1 (a;b1,b2;c; x, y) and Campe de Feriet’s function 4;4,4 [ ]
4;3,3
F x, y . We use expansion of series Appel's hypergeometric function and the method of grouping terms of smooth convergent power series. We find connections between Gaussian functions, generalized hypergeometric function 4
8 7
F (x ) and hyperbolic cosines by using new relationship.

Калит сўзлар

hypergeometric functions

Appel function

№ Муаллифнинг исми Лавозими Ташкилот номи

1 Xasanov A.. 1 Namangan state university

2 Tolasheva Y.I. 2 Namangan state university

№ Ҳавола номи

1 1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, 55, National Bureau of Standards, Washington, D. C., 1964; Reprinted by Dover Publications, New York, 1965.

2 2. A. Altin, Some expansion formulas for a class of singular partial diﬀerential equations, Proc. Amer. Math. Soc., 85(1)(1982), 42-46.

3 3. P. Appell and J. Kampeґ de Feґriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite, Gauthier - Villars, Paris, 1926.

4 4. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Duke Math. J., 98(3)(1999), 465-483.

5 5. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator II, Duke Math. J., 111(3)(2002), 561-584.

6 6. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator III, Duke Math. J., 128(1)(2005), 119-140.

7 7. L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958.

8 8. B. C. Carlson, Some extensions of Lardner’s relations between 0F3 and Bessel functions, SIAM J. Math. Anal., 1(2)(1970), 232-242.

9 9. A. Erdeґlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw- Hill Book Company, New York, Toronto and London, 1953.

10 10. A. Erdeґlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw- Hill Book Company, New York, Toronto and London, 1953.

11 11. F. I. Frankl, Selected Works in Gas Dynamics. Nauka, Moscow 1973.

12 12. A. J. Fryant, Growth and complete sequences of generalized bi-axially symmetric potentials, J. Diﬀ. Equa., 31(2)(1979), 155-164.

13 13. A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations 52(8)(2007), 673-683.

14 14. A. Hasanov, Some solutions of generalized Rassias’s equation, Intern. J. Appl. Math.Stat., 8(M07), (2007), 20- 30.

15 15. A. Hasanov, Fundamental solutions for degenerated elliptic equation with two perpendicular lines of degeneration, Intern. J. Appl. Math. Stat., 13(8)(2008), 41-49.

16 16. A. Hasanov and E. T. Karimov, Fundamental solutions for a class of three-dimensional elliptic equations with singular coeﬃcients, Appl. Math. Lett., 22(2009),1828-1832.

17 17. A. Hasanov, J. M. Rassias and M. Turaev, Fundamental solution for the generalized Elliptic Gellerstedt Equation, Book: Functional Equations, Diﬀerence Inequalities and ULAM Stability Notions, Nova Science Publishers Inc. NY, USA, 6(2010), 73-83.

18 18. A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricella Function and other multiple hypergeometric functions, Appl. Math. Lett.,19(2006), 113-121.

19 19. A. Hasanov and H. M. Srivastava, Decomposition formulas associated with the Lauricella multivariable hypergeometric functions, Comput. Math. Appl., 53(7)(2007),1119-1128.

20 20. A. Hasanov, H. M. Srivastava, and M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl., 324(2006), 955-969.

21 21. A. Hasanov and M. Turaev, Decomposition formulas for the double hypergeometric G1 and G2 Hypergeometric functions, Appl. Math. Comput., 187(1)(2007), 195-201.

22 22. Y. S. Kim, A. K. Rathie and J. Choi, Note on Srivastava’s triple hypergeometric series HA, Commun. Korean Math. Soc., 18(3)(2003), 581-586.

23 23. T. J. Lardner, Relations between 0F3 and Bessel functions, SIAM Review, 11(1969), 69-72.

24 24. T. J. Lardner and C. R. Steele, Symmetric deformations of circular cylindrical elastic shells of exponentially varying thickness, Trans. ASME Ser. E. J. Appl. Mech.,35(1968), 169-170.

25 25. G. Lohofer, Theory of an electromagnetically deviated metal sphere. 1: Absorbedpower. SIAM J. Appl. Math., 49(1989), 567-581.

26 26. P. A. McCoy, Polynomial approximation and growth of generalized axisymmetric potentials, Canad. J. Math., 31(1)(1979), 49-59.

27 27. A. W. Niukkanen, Generalized hypergeometric series arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 16(1983), 1813-1825.

28 28. A. K. Rathie and Y. S. Kim, Further results on Srivastava’s triple hypergeometric series HA and HC, Ind. J. Pure Appl. Math., 35(8), (2004), 991-1002.

29 29. M. S. Salakhitdinov and A. Hasanov, A solution of the Neumann-Dirichlet boundary value problem for generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations, 53(4)(2008), 355-364.

30 30. H. M. Srivastava, Hypergeometric functions of three variables, Ganita, 15(2)(1964), 97-108.

31 31. H. M. Srivastava, Some integrals representing hypergeometric functions, Rend. Circ. Mat. Palermo, 16(2)(1967), 99-115.

32 32. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.

33 33. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press(Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, 1985.

34 34. M. Turaev, Decomposition formulas for Srivastava’s hypergeometric function on Saran functions, Comput. Appl. Math., 233(2009), 842-846.

35 35. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edi., Cambridge University Press, Cambridge, London and New York, 1944.

36 36. A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer.Math. Soc., 63(1946), 342- 354.

37 37. A. Weinstein, Generalized axially symmetric potential theory, Bull. Amer. Math. Soc.,59(1953), 20-38.

38 38. R. J. Weinacht, Fundamental solutions for a class of singular equations, Contrib. Diﬀ.Equa., 3(1964), 43-55.

Кутилмоқда

№	Муаллифнинг исми	Лавозими	Ташкилот номи
1	Xasanov A..	1	Namangan state university
2	Tolasheva Y.I.	2	Namangan state university

№	Ҳавола номи
1	1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, 55, National Bureau of Standards, Washington, D. C., 1964; Reprinted by Dover Publications, New York, 1965.
2	2. A. Altin, Some expansion formulas for a class of singular partial diﬀerential equations, Proc. Amer. Math. Soc., 85(1)(1982), 42-46.
3	3. P. Appell and J. Kampeґ de Feґriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite, Gauthier - Villars, Paris, 1926.
4	4. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Duke Math. J., 98(3)(1999), 465-483.
5	5. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator II, Duke Math. J., 111(3)(2002), 561-584.
6	6. J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator III, Duke Math. J., 128(1)(2005), 119-140.
7	7. L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958.
8	8. B. C. Carlson, Some extensions of Lardner’s relations between 0F3 and Bessel functions, SIAM J. Math. Anal., 1(2)(1970), 232-242.
9	9. A. Erdeґlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw- Hill Book Company, New York, Toronto and London, 1953.
10	10. A. Erdeґlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw- Hill Book Company, New York, Toronto and London, 1953.
11	11. F. I. Frankl, Selected Works in Gas Dynamics. Nauka, Moscow 1973.
12	12. A. J. Fryant, Growth and complete sequences of generalized bi-axially symmetric potentials, J. Diﬀ. Equa., 31(2)(1979), 155-164.
13	13. A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations 52(8)(2007), 673-683.
14	14. A. Hasanov, Some solutions of generalized Rassias’s equation, Intern. J. Appl. Math.Stat., 8(M07), (2007), 20- 30.
15	15. A. Hasanov, Fundamental solutions for degenerated elliptic equation with two perpendicular lines of degeneration, Intern. J. Appl. Math. Stat., 13(8)(2008), 41-49.
16	16. A. Hasanov and E. T. Karimov, Fundamental solutions for a class of three-dimensional elliptic equations with singular coeﬃcients, Appl. Math. Lett., 22(2009),1828-1832.
17	17. A. Hasanov, J. M. Rassias and M. Turaev, Fundamental solution for the generalized Elliptic Gellerstedt Equation, Book: Functional Equations, Diﬀerence Inequalities and ULAM Stability Notions, Nova Science Publishers Inc. NY, USA, 6(2010), 73-83.
18	18. A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricella Function and other multiple hypergeometric functions, Appl. Math. Lett.,19(2006), 113-121.
19	19. A. Hasanov and H. M. Srivastava, Decomposition formulas associated with the Lauricella multivariable hypergeometric functions, Comput. Math. Appl., 53(7)(2007),1119-1128.
20	20. A. Hasanov, H. M. Srivastava, and M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl., 324(2006), 955-969.
21	21. A. Hasanov and M. Turaev, Decomposition formulas for the double hypergeometric G1 and G2 Hypergeometric functions, Appl. Math. Comput., 187(1)(2007), 195-201.
22	22. Y. S. Kim, A. K. Rathie and J. Choi, Note on Srivastava’s triple hypergeometric series HA, Commun. Korean Math. Soc., 18(3)(2003), 581-586.
23	23. T. J. Lardner, Relations between 0F3 and Bessel functions, SIAM Review, 11(1969), 69-72.
24	24. T. J. Lardner and C. R. Steele, Symmetric deformations of circular cylindrical elastic shells of exponentially varying thickness, Trans. ASME Ser. E. J. Appl. Mech.,35(1968), 169-170.
25	25. G. Lohofer, Theory of an electromagnetically deviated metal sphere. 1: Absorbedpower. SIAM J. Appl. Math., 49(1989), 567-581.
26	26. P. A. McCoy, Polynomial approximation and growth of generalized axisymmetric potentials, Canad. J. Math., 31(1)(1979), 49-59.
27	27. A. W. Niukkanen, Generalized hypergeometric series arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 16(1983), 1813-1825.
28	28. A. K. Rathie and Y. S. Kim, Further results on Srivastava’s triple hypergeometric series HA and HC, Ind. J. Pure Appl. Math., 35(8), (2004), 991-1002.
29	29. M. S. Salakhitdinov and A. Hasanov, A solution of the Neumann-Dirichlet boundary value problem for generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations, 53(4)(2008), 355-364.
30	30. H. M. Srivastava, Hypergeometric functions of three variables, Ganita, 15(2)(1964), 97-108.
31	31. H. M. Srivastava, Some integrals representing hypergeometric functions, Rend. Circ. Mat. Palermo, 16(2)(1967), 99-115.
32	32. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
33	33. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press(Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, 1985.
34	34. M. Turaev, Decomposition formulas for Srivastava’s hypergeometric function on Saran functions, Comput. Appl. Math., 233(2009), 842-846.
35	35. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edi., Cambridge University Press, Cambridge, London and New York, 1944.
36	36. A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer.Math. Soc., 63(1946), 342- 354.
37	37. A. Weinstein, Generalized axially symmetric potential theory, Bull. Amer. Math. Soc.,59(1953), 20-38.
38	38. R. J. Weinacht, Fundamental solutions for a class of singular equations, Contrib. Diﬀ.Equa., 3(1964), 43-55.