Product of two hypergeometric functions with power arguments. (English) Zbl 1423.33007
Summary: The hypergeometric product formula describes the product of two hypergeometric functions with arguments \(xt\) and \(yt\), respectively. The result is a power series in \(t\) whose coefficients are hypergeometric polynomials. Brychkov has extended the product formula such that in one argument the variable \(t\) can be squared. We obtain a further extension such that in both arguments the variable \(t\) can be raised to positive integer powers.
MSC:
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 33C65 | Appell, Horn and Lauricella functions |
Keywords:
generalized hypergeometric function; regularized hypergeometric function; hypergeometric polynomial; generating function; Appell sequenceSoftware:
DLMFReferences:
| [1] | Altın, A.; Erkuş, E.; Özarslan, M. A., Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct., 17, 5, 315-320 (2006), Zbl 1099.26010. MR2237492 · Zbl 1099.26010 |
| [2] | Araujo, O.; Estrada, M.; Morales, D. A.; Rada, J., The higher-order matching polynomial of a graph, Int. J. Math. Math. Sci., 10, 1565-1576 (2005), Zbl 1076.05062. MR2177862 · Zbl 1076.05062 |
| [3] | Boas, R. P.; Buck, R. C., Polynomial Expansions of Analytic Functions (1964), Springer-Verlag, Zbl 0116.28105. MR0162914 · Zbl 0116.28105 |
| [4] | Brafman, F., Generating functions of Jacobi and related polynomials, Proc. Amer. Math. Soc., 2, 942-949 (1951), Zbl 0044.07602. MR0045875 · Zbl 0044.07602 |
| [5] | Brychkov, Yu. A., On power series of products of special functions, Appl. Math. Lett., 24, 8, 1374-1378 (2011), Zbl 1233.33001. MR2793636 · Zbl 1233.33001 |
| [6] | Carlson, B. C., Polynomials satisfying a binomial theorem, J. Math. Anal. Appl., 32, 543-558 (1970), Zbl 0204.38301. MR0271405 · Zbl 0204.38301 |
| [7] | Chan, W.-Ch. C.; Chyan, Ch.-J.; Srivastava, H. M., The Lagrange polynomials in several variables, Integral Transforms Spec. Funct., 12, 2, 139-148 (2001), Zbl 1057.33003. MR1867910 · Zbl 1057.33003 |
| [8] | Dattoli, G., Generalized polynomials, operational identities and their applications, J. Comput. Appl. Math., 118, 1-2, 111-123 (2000), Zbl 1016.33012. MR1765943 · Zbl 1016.33012 |
| [9] | Dattoli, G.; Torre, A.; Lorenzutta, S., Operational identities and properties of ordinary and generalized special functions, J. Math. Anal. Appl., 236, 2, 399-414 (1999), Zbl 0943.33005. MR1704591 · Zbl 0943.33005 |
| [10] | Driver, K. A.; Love, A. D., Products of hypergeometric functions and the zeros of \({}_4F_3\) polynomials, Numer. Algorithms, 26, 1, 1-9 (2001), Zbl 0983.65022. MR1827314 · Zbl 0983.65022 |
| [11] | Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions. Vol. II (1981), Krieger Publ., Zbl 0505.33001. MR698780 · Zbl 0052.29502 |
| [12] | Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions. Vol. III (1981), Krieger Publ., Zbl 0542.33002. MR698781 · Zbl 0064.06302 |
| [13] | Erkuş-Duman, E., Matrix extensions of polynomials in several variables, Util. Math., 85, 161-180 (2011), Zbl 1293.15009. MR2840794 · Zbl 1293.15009 |
| [14] | Erkuş, E.; Altın, A., A note on the Lagrange polynomials in several variables, J. Math. Anal. Appl., 310, 1, 338-341 (2005), Zbl 1082.33007. MR2160694 · Zbl 1082.33007 |
| [15] | Erkuş-Duman, E.; Altın, A.; Aktaş, R., Miscellaneous properties of some multivariable polynomials, Math. Comput. Modelling, 54, 9-10, 1875-1885 (2011), Zbl 1235.33012. MR2834594 · Zbl 1235.33012 |
| [16] | Erkuş, E.; Srivastava, H. M., A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct., 17, 4, 267-273 (2006), Zbl 1098.33016. MR2236211 · Zbl 1098.33016 |
| [17] | Gaboury, S.; Tremblay, R., Generalized multiple integral representations for a large family of polynomials with applications, Appl. Math. Comput., 261, 39-47 (2015), Zbl 06890723. MR3345255 · Zbl 1410.33029 |
| [18] | Gaboury, S.; Tremblay, R.; Özarslan, M. A., Some bilateral generating functions involving the Erkuş-Srivastava polynomials and some general classes of multivariable polynomials, Tamkang J. Math., 45, 4, 341-356 (2014), Zbl 1320.30009. MR3302308 · Zbl 1320.30009 |
| [19] | Gould, H. W.; Hopper, A. T., Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29, 51-63 (1962), Zbl 0108.06504. MR0132853 · Zbl 0108.06504 |
| [20] | Grinshpan, A. Z., Generalized hypergeometric functions: product identities and weighted norm inequalities, Ramanujan J., 31, 1-2, 53-66 (2013), Zbl 1275.33011. MR3048654 · Zbl 1275.33011 |
| [21] | Horozov, E., \(d\)-orthogonal analogs of classical orthogonal polynomials, SIGMA Symmetry Integrability Geom. Methods Appl., 14 (2018), Zbl 06910009. MR3817776 · Zbl 1393.33012 |
| [22] | Khan, S.; Al-Gonah, A. A., Operational methods and Laguerre-Gould Hopper polynomials, Appl. Math. Comput., 218, 19, 9930-9942 (2012), Zbl 1250.33012. MR2916175 · Zbl 1250.33012 |
| [23] | Khan, S.; Raza, N.; Ali, M., Finding mixed families of special polynomials associated with Appell sequences, J. Math. Anal. Appl., 447, 1, 398-418 (2017), Zbl 1355.33016. MR3566479 · Zbl 1355.33016 |
| [24] | Khan, S.; Walid Al-Saad, M., Summation formulae for Gould-Hopper generalized Hermite polynomials, Comput. Math. Appl., 61, 6, 1536-1541 (2011), Zbl 1217.33020. MR2775919 · Zbl 1217.33020 |
| [25] | Lin, S.-D.; Chao, Y.-S.; Srivastava, H. M., Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl., 294, 2, 399-411 (2004), Zbl 1048.33012. MR2061333 · Zbl 1048.33012 |
| [26] | Lin, S.-D.; Liu, S.-J.; Lu, H.-C.; Srivastava, H. M., Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials, Appl. Math. Comput., 218, 4, 1330-1341 (2011), Zbl 1242.33022. MR2831641 · Zbl 1242.33022 |
| [27] | Lin, S.-D.; Liu, S.-J.; Lu, H.-C.; Srivastava, H. M., Linearization relations for the generalized Bedient polynomials of the first and second kinds via their integral representations, Czechoslovak Math. J., 63(138), 4, 969-987 (2013), Zbl 1313.33008. MR3165508 · Zbl 1313.33008 |
| [28] | Lin, S.-D.; Liu, S.-J.; Srivastava, H. M., Some families of hypergeometric polynomials and associated multiple integral representations, Integral Transforms Spec. Funct., 22, 6, 403-414 (2011), Zbl 1258.33007. MR2811972 · Zbl 1258.33007 |
| [29] | Lin, S.-D.; Srivastava, H. M.; Wang, P.-Y., Some families of hypergeometric transformations and generating relations, Math. Comput. Modelling, 36, 4-5, 445-459 (2002), Zbl 1042.33005. MR1928601 · Zbl 1042.33005 |
| [30] | Liu, S.-J.; Lin, S.-D.; Srivastava, H. M.; Wong, M.-M., Bilateral generating functions for the Erkuş-Srivastava polynomials and the generalized Lauricella functions, Appl. Math. Comput., 218, 15, 7685-7693 (2012), Zbl 1241.33015. MR2900101 · Zbl 1241.33015 |
| [31] | Luke, Y. L., The Special Functions and Their Approximations. Vol. I (1969), Academic Press, Zbl 0193.01701. MR0249668 · Zbl 0193.01701 |
| [32] | Mathai, A. M.; Saxena, R. K., Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences (1973), Springer-Verlag: Springer-Verlag Berlin-New York, Zbl 0272.33001. MR0463524 · Zbl 0272.33001 |
| [33] | (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge Univ. Press), Zbl 1198.00002. MR2723248 · Zbl 1198.00002 |
| [34] | Rosenbloom, P. C.; Widder, D. V., Expansions in terms of heat polynomials and associated functions, Trans. Amer. Math. Soc., 92, 220-266 (1959), Zbl 0086.27203. MR0107118 · Zbl 0086.27203 |
| [35] | Srivastava, H. M., Finite summation formulas associated with a class of generalized hypergeometric polynomials, J. Math. Anal. Appl., 23, 453-458 (1968), Zbl 0162.36902. MR0228721 · Zbl 0162.36902 |
| [36] | Srivastava, H. M., On \(q\)-generating functions and certain formulas of David Zeitlin, Illinois J. Math., 15, 64-72 (1971), Zbl 0205.35902. MR0279347 · Zbl 0205.35902 |
| [37] | Srivastava, H. M., A note on certain identities involving generalized hypergeometric series, Indag. Math., 41, 2, 191-201 (1979), Zbl 0387.33003. MR535567 · Zbl 0387.33003 |
| [38] | Srivastava, H. M., Some families of generating functions associated with the Stirling numbers of the second kind, J. Math. Anal. Appl., 251, 2, 752-769 (2000), Zbl 0972.33005. MR1794769 · Zbl 0972.33005 |
| [39] | Srivastava, H. M.; Lin, S.-D.; Liu, S.-J.; Lu, H.-C., Integral representations for the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and the generalized Bessel polynomials, Russ. J. Math. Phys., 19, 1, 121-130 (2012), Zbl 1259.33013. MR2892608 · Zbl 1259.33013 |
| [40] | Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions (1984), John Wiley & Sons, Zbl 0535.33001. MR750112 · Zbl 0535.33001 |
| [41] | Srivastava, H. M.; Özarslan, M. A.; Kaanoğlu, C., Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Russ. J. Math. Phys., 20, 1, 110-120 (2013), Zbl 1292.11043. MR3071909 · Zbl 1292.11043 |
| [42] | Widder, D. V., The Heat Equation (1975), Academic Press, Zbl 0322.35041. MR0466967 · Zbl 0322.35041 |
| [43] | Yasmin, G., Some properties of Legendre-Gould Hopper polynomials and operational methods, J. Math. Anal. Appl., 413, 1, 84-99 (2014), Zbl 1308.33008. MR3153570 · Zbl 1308.33008 |
| [44] | Zeitlin, D., On generating functions and a formula of Chaudhuri, Amer. Math. Monthly, 74, 1056-1062 (1967), Zbl 0153.39401. MR0222361 · Zbl 0153.39401 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
