Acknowledgement
The authors would like to express their deep-felt thanks for the reviewers' encouraging comments. The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R111A1A01052440).
References
- M. Abbas and S. Bouroubi, On new identities for Bells polynomial, Discrete Math., 293(13) (2005), 5-10. doi:10.1016/j.disc.2004.08.023.
- M. Abramowitz, I.A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, ninth printing, National Bureau of Standards, Washington, D.C., 1972; Reprint of the 1972 Edition, Dover Publications, Inc., New York, 1992.
- G.E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.
- L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985.
- L.C. Andrews, Special Functions of Mathematics for Engineers, Reprint of the 1992 Second Edition, SPIE Optical Engineering Press, Bellingham, W.A., Oxford University Press, Oxford, 1998.
- W.N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc. s2-28(1) (1928), 242-254. doi:10.1112/plms/s2-28.1.242.
- W.N. Bailey, Transformations of generalized hypergeometric series, Proc. London Math. Soc. s2-29(1) (1929), 495-516. https://doi.org/10.1112/plms/s2-29.1.495
- A.H. Bhat, M.I. Qureshi and J. Majid, Hypergeometric forms of certain composite functions involving arcsine(x) using Maclaurin series and their applications, Jnanabha 50(2) (2020), 139-145.
- E.T. Bell, Partition polynomials, Ann. Math., 29(1/4) (1927-1928), 38-46. doi:10.2307/1967979.
- Yury A. Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylor & Fancis Group, Boca Raton, London, New York, 2008.
- C.-P. Chen and J. Choi, Asymptotic expansions for the constants of Landau and Lebesgue, Adv. Math., 254 (2014), 622-641. http://dx.doi.org/10.1016/j.aim. 2013.12.021.
- J. Choi, Certain applications of generalized Kummer's summation formulas for 2F1, Symmetry 13 (2021), Article ID 1538. https://doi.org/10.3390/sym13081538.
- J. Choi, M. I. Qureshi, A. H. Bhat and J. Majid, Reduction formulas for generalized hypergeometric series associated with new sequences and applications, Fractal Fract., 5 (2021), Article ID 150. https://doi.org/10.3390/fractalfract5040150.
- L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Dordrecht, Holland / Boston, U.S.: Reidel Publishing Company, 1974.
- A.D.D. Craik, Prehistory of Faa di Brunos formula, Amer. Math. Monthly, 112(2) (2005), 217-234. http://www.jstor.org/stable/30037410 (accessed on 2021-05-02).
- D. Cvijovic, New identities for the partial Bell polynomials, Appl. Math. Lett., 24(9) (2011), 1544-1547. doi:10.1016/j.aml.2011.03.043.
- A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
- I. Gessel, Finding identities with the WZ method, J. Symbolic Comput., 20(5/6) (1995), 537-566. https://doi.org/10.1006/jsco.1995.1064.
- I. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal., 13(2) (1982), 295-308. https://doi.org/10.1137/0513021.
- I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 6th edition, Academic Press, San Diego, San Francisco, New York, Boston, London, Sydney, Tokyo, 2000.
- W.P. Johnson, The curious history of Fa di Brunos formula, Amer. Math. Monthly, 109(3) (2002), 217-234. http://www.jstor.org/stable/2695352 (accessed on 2021-05-02). https://doi.org/10.2307/2695352
- C. Krattenthaler and K. Srinivasa Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math., 160 (2003), 159-173. https://doi.org/10.1016/S0377-0427(03)00629-0.
- N.N. Lebedev, Special Functions and Their Applications, Revised English Edition (Translated and edited by Richard A. Silverman), Dover Publications, Inc., New York, 1972.
- W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third enlarged Edition, Springer-Verlag, New York, 1966.
- S. Noschese and P.E. Ricci, Differentiation of multivariable composite functions and Bell polynomials, J. Comput. Anal. Appl., 5(3) (2003), 333-340. doi:10.1023/A:1023227705558.
- F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark (Editors), NIST Handbook of Mathematical Functions, NIST and Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi, Dubai, Tokyo, 2010.
- P. Paule, A proof of a conjecture of Knuth, Exp. Math., 5(2) (1996), 83-89. https://doi.org/10.1080/10586458.1996.10504579.
- A.P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, Integrals and Series, More Special Functions, Vol. 3, Nauka Moscow, 1986 (in Russian); (Translated from the Russian by G. G. Gould), Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo, Melbourne, 1990.
- M.I. Qureshi, S.H. Malik and J. Ara, Hypergeometric forms of some mathematical functions via differential equation approach, Jnanabha 50(2) (2020), 153-159.
- M.I. Qureshi, J. Majid and A.H. Bhat, Hypergeometric forms of some composite functions containing arccosine(x) using Maclaurins Expansion, South East Asian J. Math. Math. Sci., 16(3) (2020), 83-96.
- M.I. Qureshi, S.H. Malik and T.R. Shah, Hypergeometric forms of some functions involving arcsine(x) using differential equation approach, South East Asian J. Math. Math. Sci., 16(2) (2020), 79-88.
- M.I. Qureshi, S.H. Malik and T.R. Shah, Hypergeometric representations of some mathematical functions via Maclaurin series, Jnanabha 50(1) (2020), 179-188.
- E.D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
- L.J. Slater, Generalized Hypergeometric Functions, Cambridge at the University Press, London, New York, 1966.
- H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
- H.M. Srivastava and P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
- H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
- G. Szego, Orthogonal Polynomials, Vol. XXIII, Amer. Math. Soc., Providence, Rhode Island, Colloquium publ., New York, 1939.
- F.J.W. Whipple, Well-poised series and other generalized hypergeometrtc series, Proc. London Math. Soc., s2-25(1) (1926), 525-544. https://doi.org/10.1112/plms/s2-25.1.525
- F.J.W. Whipple, Some transformations of generalized hypergeometric series, Proc. London Math. Soc., s2-26 (1927), 257-272. https://doi.org/10.1112/plms/s2-26.1.257
- H.S. Wilf and D. Zeiberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., 3 (1990), 147-158. https://doi.org/10.1090/S0894-0347-1990-1007910-7
- https://en.wikipedia.org/wiki/Fa_di_Brunos_formula
- http://en.wikipedia.org/wiki/Bell_polynomials