Refine your search

Collections

Co-Authors

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z **All**

### Rai, Prakriti

- Some Properties of Extended Hypergeometric Function and its Transformations

Abstract Views :288 |
PDF Views:2

1 Department of Mathematics, Amity Institute of Applied Sciences Amity University, Noida, IN

#### Authors

**Affiliations**

1 Department of Mathematics, Amity Institute of Applied Sciences Amity University, Noida, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 85, No 3-4 (2018), Pagination: 305-312#### Abstract

There emerges different extended versions of Beta function and hypergeometric functions containing extra parameters. We obtain some properties of certain functions like extended Generalized Gauss hypergeometric functions, extended Confluent hypergeometric functions including transformation formulas, Mellin transformation for the generalized extended Gauss hypergeometric function in one, two and more variables.#### Keywords

Extended Gamma and Extended Beta Functions, Extended Gauss Hypergeometric Functions, Extended Confluent Hypergeometric Functions, Mellin Transforms.#### References

- L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, New York, McMillan (1985).
- M.A. Chaudhry and S.M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), 99-124.
- M.A. Chaudhry, A. Qadir M. Raque and S.M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math. 78 (1997), 19-32.
- L. Minjie, A Class of Extended Hypergeometric Functions and Its Applications, arXiv:1302.2307v1, [math.CA]. 2013.
- E. Ozergin, M.A. Ozarslan and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610.
- M.A. Ozarslan and E. Ozergin, Some generating relations for extended hypergeometric function via generalized fractional derivative operator, Math. Comput. Model., 52 (2010), 1825-1833.
- H.M. Srivastava, R.K. Parmar and P. Chopra, A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions, Axioms 2012, I, 238-258.

- Generalized Hermite-Based Apostol-Bernoulli, Euler, Genocchi Polynomials and their Relations

Abstract Views :318 |
PDF Views:1

1 Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, IN

#### Authors

**Affiliations**

1 Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 9-21#### Abstract

In this paper, we have generalized Apostol-Hermite-Bernoullli polynomials, Apostol-Hermite-Euler polynomials and Apostol-Hermite-Genocchi polynomials. We have shown that there is an intimate connection between these polynomials and derived some implicit summation formulae by applying the generating functions#### Keywords

2010 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

Apostol-Hermite-Bernoullli Polynomials, Apostol-Hermite-Euler Polynomials and Apostol-Hermite-Genocchi Polynomials, Summation Formulae, Symmetric Identities.

#### References

- T. M. Apostol, On the Lerch Zeta function, Pacific J. Math., 1(1951), 161-167.
- E. T. Bell, Exponential polynomials, Ann. Math., 35(1934), 258-277.
- G. Dattoli, B. Germano and P. E. Ricci, Hermite polynomials with more than two variables and associated bi-orthogonal functions, Integral Transforms and Special Functions, 20(1) (2009), 17-22.
- Q. M. Luo, Fourier expansions and integral representations for the Apostol-Bernoulli and ApostolEuler polynomials, Math. of Comp. 78 (2009), 2193-2208.
- Q. M. Luo, Fourier expansions and integral representations for the Genocchi polynomials, J. Integer Seq. 12 (2009), 1-9.
- Q. M. Luo, q-extension for the Apostol-Genocchi polynomials, Gen. Math. 17 (2009), 113-125.
- Q. M. Luo, Some formulas for the Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments, Applicable Analysis and Discrete Mathematics 3(2) (2009), 336-346.
- Q. M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transform and Special Functions, 20(5-6) (2009), 377-391.
- Q. M. Luo, Extension for the Genocchi polynomials and its Fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291-310.
- Q. M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308(2005), 290-302.
- Q. M. Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Computers and Mathematics with Applications, 51(3-4) (2006), 631-642.
- Q. M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Applied Math. & Comput. 217(12) (2011), 5702-5728.
- M. A. O¨ zarslan, Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials, Adv. Diff. Eq. (2013), 116, DOI: 10.1186/1687-1847-2013-116.
- M. A. Pathan and W. A. Khan, Some new classes of generalized Hermite-based Apostol-Euler and Apostol-Genocchi polynomials, Fasciculi Mathematici, 55(2015), 153-170, DOI: 10.1515/fascmath2015-0020.
- E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
- H. M. Srivastava and H. L. Manocha, Atreatise on Generating Functions, Halsted, New York, 1984.
- H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), 77-84.
- R. Tremblay, S. Gaboury and B. J. Fugere, Some new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials, Int. J. Math and Math. Sci. (2012), DOI:10.1155/2012/182785.
- R. Tremblay, S. Gaboury and B. J. Fugere, A further generalization of Apostol-Bernoulli polynomials and related polynomials, Honam Math. J. 34 (2012), 311-326.

- A Generalization of Class of Humbert - Hermite Polynomials

Abstract Views :172 |
PDF Views:0

1 Department of Mathematics, Institute of Applied Sciences, Amity University, Noida, IN

2 Department of Mathematics, Siddharth University, Kapilvastu, India Siddharth Nagar, IN

#### Authors

**Affiliations**

1 Department of Mathematics, Institute of Applied Sciences, Amity University, Noida, IN

2 Department of Mathematics, Siddharth University, Kapilvastu, India Siddharth Nagar, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 89, No 3-4 (2022), Pagination: 227-236#### Abstract

A generalization of Humbert-Hermite polynomials is defined in this paper. Moreover, several generalizations of Hermite-Gegenbauer polynomials, Hermite-Legendre and Hermite-Chebyshev polynomials are established.

#### Keywords

Hermite Polynomials, Humbert Polynomials, Gegenbauer Polynomials, Legendre Polynomials, Chebyshev Polynomials, Hypergeometric Function.#### References

- E. T. Bell, Exponential polynomials, Ann. of Math., 35(1934), 258-277.
- A. Chaturvedi, and Rai, P., Generalized Hermite-based Apostol-Bernoulli, Euler, Genocchi polynomials and their relations, Journal of Indian Mathematical Society, 87(1-2)(2020), 9-21.
- J. Choi, Notes on formal manipulations ofdouble series, Commun. Korean Math. Soc., 18(4) (2003), 781-789.
- Dattoli G., Germano B. and Ricci P. E., Hermite polynomials with more than two variables and associated bi-orthogonal functions, Integral Transforms and Special Functions, 20(1) (2009), 17-22.
- G. Dattoli, S. Lorenzutta and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rendiconti di Mathematica, 19(1999), 385–391.
- G. B. Djordjevi´c, A generalization of Gegenbauer polynomial with two variables, Indian J. Pure Appl. Math., (To appear).
- T. Kim, j. Choi, Y. H. Kim and C. S. Ryoo, On q-Bernstein and q-Hermite polynomials, Proc. Jangjeon Math. Soc., 14(A202) (2011), 215–221.
- G. V. Milovanovi´c and G. B. Djordjevi´c, On some properties of Humberts polynomials-I, Fibonacci Quart., 25(1987), 356–360.
- M. A. Pathan and M. A. Khan, On polynomials associated with Humberts polynomials, Publ. Inst. Math., (Beograd), 62(76) (1997), 53–62.
- M. A. Pathan and N. U. Khan, A uni?ed presentation of a class of generalized Humbert Polynomials in two variables, ROMAI J., 11(2) (2015), 185–199.
- M. A. Pathan and W. Khan, On a class of Humbert-Hermite polynomials, Novi Sad J. Math., 51(1) (2021), 1–11.
- Y. Simsek and M. Acikgoz, A new generating function of (q?)Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, 2010 (2010), Article ID 769095, 12 Pages.

- Multi-Indexed Whittaker Function and its Properties

Abstract Views :125 |
PDF Views:0

1 Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, IN

2 Department of Mathematics, Siddharth University, Kapilvastu, IN

#### Authors

**Affiliations**

1 Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, IN

2 Department of Mathematics, Siddharth University, Kapilvastu, IN

#### Source

The Journal of the Indian Mathematical Society, Vol 90, No 1-2 (2023), Pagination: 115-124#### Abstract

In this paper, we have introduced the multi-indexed Whittaker function (3m-parameter) by using the extended confluent hypergeometric function which is defined in terms of multi-indexed (3m-parameter) Mittag-Leffler function. We derive some properties of multi-indexed (3m-parameter) Whittaker function such as its integral representations, derivative formula and Hankel transform.#### Keywords

Extended Beta Function, Gauss Hypergeometric Function, Confluent Hypergeometric Function, Multi-Index Mittag-Leffler Function, Whittaker Function and Extended Whittaker Function.#### References

- M. Ali, M. Ghayasuddin, W. A. Khan and K. S. Nisar, A novel Kind of the multi-index Beta, Gauss and confluent hypergeometric functions, J. Math. Computer Sci., 23 (2021), 145–154.
- M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589–602.
- M. A. Chaudhry, A. Qadir, M. Rafique and S. M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math., 78 (1997), 19–32.
- A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Table of Integral Transforms, Vol. 1, McGraw-Hill, New York, 1954.
- M. Ghayasuddin, N. U. Khan and M. Ali, A study of extended beta, Gauss and confluent hypergeometric functions, Inter. J. Appl. Math., 33 (2020), 01-13.
- V. S. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput. Math. Appl., 59 (2010), 1885–1895.
- V. S. Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math., 118 (2000), 241–259.
- G. M. Mittag-Leffler, Sur la representation analytique d´une branche uniforme d´une fonction monogene, Acta Math., 29 (1905), 101–182.
- D. K. Nagar, R. A. M. V´asquez and A. K. Gupta, Properties of the extended Whittaker function, Progr. Appl. Math., 6(2) (2013), 70–80.
- J. Paneva-Konovska, Multi-index(3m-parametric) Mittag-Leffler functions and fractional calculus, C. R. Acad. Bulgare Sci., 64 (2011), 1089–1098.
- J. Paneva-Konovska, From Bessel to multi-index Mittag-Leffler functions: Enumerable families,series in them and convergence,World Scientific Publishing, London, 2016.
- J. Paneva-Konovska, A survey on Bessel type functions as multi-index Mittag-Leffler functions: Differential and integral relations, Int. J. Appl. Math., 32 (2019), 357–380.
- E. D. Rainville, Special functions, The Macmillan Company, New York,1960, Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
- M. Shadab, S. Jabee, J. Choi, An extended beta function and its application, Far East J. Math. Sci., 103 (2018), 235–251.
- H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Eills Horwood Limited; Chichester ), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
- E. T. Whittaker, An expression of certain known functions as generalized hypergeometric functions, Bull. Amer. Math. Soc., 10(3) (1903), 125–134.