Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 52 https://internationalpubls.com Generalization of Classical Summation Relation of Certain Appell’s Double Hypergeometric Functions Associated with Theory of Approximation Madhav Prasad Poudel1,Narayan Prasad Pahari2, Suresh Kumar Sahani*3, Ganesh Bahadur Basnet4, & Resham Prasad Poudel5 1School of Engineering, Pokhara University, Pokhara-30, Kaski, Nepal 1pdmadav@gmail.com 2Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal 2nppahari@gmail.com *3Department of Mathematics, Janakpur Campus, Tribhuvan University, Janakpurdham, Nepal 3sureshkumarsahani35@gmail.com 4,5Department of Mathematics, Tribhuvan University, Tri-Chandra Campus, Kathmandu, Nepal 4gbbmath@gmail.com, and 5reshamprdpaudel@gmail.com Corresponding author: Suresh Kumar Sahani*3 Corresponding author Email: sureshkumarsahani35@gmail.com Article History: Received: 07-09-2023 Revised: 18-10-2023 Accepted: 12-11-2023 Abstract: In this research note, we have obtained some new classical summation relations of certain Appell’s double hypergeometric functions associated with theory of approximation. This novel relation include, as special case, a set of well-known results. It studies the famous works of the authors [1], [2], [14], [15], [33] –[38]. Keywords: Hypergeometric functions, Srivastava’s triple hypergeometric functions, Pochhammer symbol, summation, and etc. MSC: 33C05, 33C90, 33C65. 1. Introduction Let ( ) ( ) ( )g g    + =  ( )( ) ( ) ( ) 0 1 2 ... 1 , 1g    = + + + − = (1) and g is a positive number and ( ) ( ) ( ) 1 , 0 1g g g    − − = −  − (2) In 1926, Appell’s (see [1], [3], [4]) defined the following function ( )1 1 , , ; ; ,F a b    = ( ) ( ) ( ) ( ) 1 0 0 . . ! ! f g f g g g f g f g a b f g       + = = +  (3) In 1833, Lauricella (see [2]) defined n -ple hypergeometric function Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 53 https://internationalpubls.com ( )1 2 1 2, , ,..., ; ; , ,...,D g gF a a a     = ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 1 2 1... 1 ,..., 0 1... ... . ... ! ! g g g g g ffgf f f f f g f f gf f f aa f f      + + + = + + +  (4) which corresponds to equation (3) where 2g = . In 1964, Srivastava (see [3], [33], [35]) defined AH by the triple series in the following way  , , ; , ; , ,AH r s a b c   = ( ) ( ) ( ) ( ) ( ), , 0 . . . ! ! ! f g f f g g f g g g a b c r s f g    + + + = +  (5) whose region of convergence is , , , 1a d b e c j d e j j   + + = + which is the generalization of [1]. The general triple hypergeometric function series ( )3 F , defined as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 11 1 11 3 1 11 1 11 : : ; ; : : : ; , , : : ; ; : : : ; F x y z r s s s t t t               ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 11 1 11 , , 0 f g f g g f f g f g f g g f r s s s     + + + + + = + + + + +  ( ) ( ) ( ) ( ) ( ) ( ) 1 11 1 11 . . . ! ! ! f g f g f g a b c f gt t t    (6) It is clear that ( ) ( ) 1 A f f    = = ( ) ( )1 A f     + =  =   (7) ( ) ( ) ( )1 B g g     =  + =   (8) and ( ) ( ) ( )1 C       + =  =   (9) where, A of the ( ) parameters, B of the ( ) parameters, and C of the ( ) parameters. The region of convergence of the triple series (6) (see [5] and [6]) are follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 54 https://internationalpubls.com 11 11 1 1 1 1 1 11 11 1 11 11 1 11 1 11 , , A B , , , , , , , , , , , A B B C D E E F A B B C D E E F B C D E E F A B C D E F A A B B + + +  + + + + + +  + + + + + +  + + + 1 11 1 1 1 11, , , , ,C C E E F and F are positive integers and , , 1and    but 11 11 1,A B B C D E E F+ + + = + + + + 1 11 11 1 11 11 1,A B B C D E E F+ + + = + + + + and 1 1 1 1 1A B B C D E E F+ + + = + + + + In 1941, a Burchnell and Chaundy (see [5] and [6]) first time to describe the following function; ( ) ( ) ( ) ( ) ( ) ( ) : : ;: ; , : ; ;: ; rA B D F a b w u vF G H        =            , 0 ! ! f g f g f g f g f g f g a a w u v f g    + = +  (10) where ( ) and ( ) f g  + are the sequences of A parameters 1 2, ,..., A   and the product ( ) ( ) ( )1 2 ... Af g f g f g    + + + respectively. Many works dealing with generalized hypergeometris function (see [24, 25]) and Srivastava’s triple hypergeometric functions and their associated properties has been done (see [7]-[34]). In this research note we obtain a Srivastava triple hypergeometric series ( )3 F instead of Kampe’de Feriet’s double hypergeometric series : ; : ; f g F i j k       . In this research paper we prove the following theorems ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 3 1 1 0 : : ; ; : , ; , ; ; , , ! : : ; ; : : ; f g g u g r f g uf F a b c g v s v w     =  − − − +−   =  −    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 11 1 1 : , ; , 1 , ;: ;1 , 1 : ; : , ; 1 , ; f f f D Ef f ff a s r u f f s f uG A C E G b F C aH B F D Hs v f w r f v       −  + − − −+ + +  −  + + + − −   where 0x  (11) Proof: We consider the series Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 55 https://internationalpubls.com ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 3 1 1 0 :: ; ; : , ; , ; ; , , ! :: ; ; : ; ; ; g g u g r f g uf F a b c g v s v w        =  − − − +−  =  −    ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 0 0! ! g g uf c g v w      = = − =  . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 0 0 . . . . ! ! j sg f g j s j sj s s j s j s j s j sj s s r u u a b f g f j ss v v     − + + = = + + + + − − + +  (12) with the help of finite triple series identity of Srivastava [34],we get ( ) ( ) 0 0 0 0 0 0 , , , , , , f g f g f f k f j k g j k k j g A f g j k A f g j j k − − − − = = = = = = = +   (13) and Pochhammer’s identities are follows: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 ! 1 ! fA g f f k f k kD f k f k k k s f k r r k f g k g k g     − − = − − −  = +  − = − −  +  − − = −   (14) Also, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 11 1 1 1 1 1 0 0 ! ! ! kf k kk k k k kk k u uu bc f kv w v v         = = + + − = + +   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 0 1 ! ! jf k f j k j j gj f j k g j g j jj k k r a j f j kk k s     = − − − − = = + + + − − −+ + +   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 1 1 0 0 0 1 ! ! ! k jf f k k j jkk k jk k j k k j jk k j ru u c b k jv w v s      − ++ ++ = = = ++ + + = −   ( )1 g f j k g − −  −     (15) now, using the following identity ( ) 0 1 0 1 0 1 t g g t for t g for t= =   − =       (16) in equation (15), without loss of generality we may assume that all terms or all values of j summation vanishing and then put j f k= − in also equation (15) we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 56 https://internationalpubls.com ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 1 1 0 0 ! 1 ! ! ! f k kf f f kk fk k f f kk f r a fu u c b kv w v j s f k      −  −++ = = −+ + = − −  (17) Thus, ( ) ( ) ( )1 ... 1! Df k f k f k r r r − − − = ( )( ) ( ) ( ) ( ) ( ) 1 1 11 ... 1 1 kk D f Dk k rr r f r f −− = − − − − ( ) ( ) ( ) 1 1 kD k r r f − = − − (18) Thus, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 , 1 1 1 0 0 1 1 !1 k D Ef f f k k kf k k f f k kf k r u f f s f u ks v f w r f v          + = = + + − − − − = + − −  ( ) ! k cb a       and our result follows. For cases reducibility: Put 0b= and 1 1 0A B G H= = = = in our main theorem i.e. in equation (11), we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 0 : , ; ;: 1; , ! : ;F : ; ; g g g rf A D C F a c g B E s w     =  −− +        ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 , ;1 , ; f ff ff r a fF A C B F c s f w       +  = + +  +   (19) Put 0c = and 1 1 1 1 0A B G H= = = = in our main theorem i.e., in equation (11), we get ( ) ( ) ( ) ( ) ( ) ( ) ( )0 : , ; , ;: 1; 1 , : ; ;: ;! f g g f g r f g uA D G F a b s vB E Fg  = − − − + + +      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 1 , ; 1 1 1 , ; f D Ef f f f r a f s f uF b E G D H r f vs a   −− − −   + + + −   − −    (20) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 57 https://internationalpubls.com Without loss of generality we may assume 1A E H= = = and 0B D G= = = in above equation (20), making suitable arrangements of parameters and also using the arbitrary definition of Jacobi polynomials, we get ( ) ( ) ( )  2 0 1 , , ;1 ,1 ; , ! ! g f g g f F r g g f a b g f g      − = − − + − − + + −  ( ) ( ) ( ) ( ) ( ), 1 1 f ff f f r b a a b L b a    − −  = +   + +  (21) where 2F is Appell’s polynomial of second kind, given by   ( ) ( ) ( ) ( ) ( ) 2 , 0 ; , ; ; , ! ! f g f g f g f g g g a b F r a b r s f g        + = =  which is the result of [28]. Similarly, putting 0A B= = in equation (20), we get ( ) ( ) ( ) ( ) ( )0 , ; , ; 1 1 ; ;! f g F F E H g f g r f g u D a G b s vg= − − − +    + +         ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 1 , ; 1 1 1 , ; f D Ef F D f r a f s f u b E G H r f vs a −− − −  = + + + −  − −  (22) which is the result of [27]. Putting 1D E H G= = = = in (22), we get ( ) 0 , ; , ; 2 1 2 1 ; ;! g g f F g F f g a b rg     = − − − +             ( ) ( ) ( ) ( ) , 1 ; ; 3 2 1 , ; f f f a f fF b f r a     − − −  =   − −  (23) which is the result of [29]. Putting 1 1 1A A B C E H= = = = = = and 1 1 0B D F G G H= = = = = = in our main theorem i.e., in equation (11), arranging parameters and using the definition of Jacobi polynomial, we get ( ) ( )3 0 :: ; ; ; ; ; ; , , :: ; ; ;1 b;1 v; ;! f g g f r g f g F a b c g   = − − − − − +    − − − + + −   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),v ! 1 , ; 2 1 ;1 1 f ff f b F f f f f A r F r f r b a b a c L fv b b a − +  −  = +    ++ + +   (24) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 58 https://internationalpubls.com which is the result of [30]. Putting 1 1 1A B C A E H= = = = = = and 1 1 0B G H D G F= = = = = = , replacing ,g f into ,f g in our main theorem and making suitable arrangement of variables of parameters, we get ( ) ( )3 0 :: ; ; ; ; ; ; ! :: ; ; ;1 ;1 ; , , , 1 1 1 g f f f f gg F a a c f s v a a c     = − − − − − −   −  − − − + + − + + +   ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 , ; 2 1 ;1 1 1 1 1 g g g g g g g g s v F ga c gs v s v c c       + + + −   =     ++ + + + + +    (25) which is the result of [31]. Also, putting 0c = and c a= in (25), we get ( )3 :: ; ; ; ; ; ; :: ; ; ; ; ; ; , , 1 1 1 u w r F a b c u s c c c      − − − −    =  − − − − + + +  ( ) ( )3 , :: ; ; ; r; ;w;1 , , :: ; ; : ; ; ; c F b c a a r s u    − − − − + − −  − − − −  (26) which is the result of [31]. Putting 0A E H= = = and 1B D G= = = in equation (20), we get ( ) ( )3 0 , ; , ; , , ! f g g f F g f g r u a b g  = − − + ( ) ( ) , ; 2 1 1 ; f f f r a F f u b r f a − − =   − −  (27) Where 3F is Appell’s polynomial of third kind given by   ( ) ( ) ( ) ( ) ( ) 3 , 0 , ; , ; ; , ! ! f g f g f g f g f g r a b F r s a b s f g        = + =  Putting 0, 1A B D G E H= = = = = − in equation (20), applying the definitions of Laguerrl polynomials and Jacobi polynomials, we get ( ) ( ) ( ) ( ) ( ) ( ) 0 1 f g s v g f g g g v f M a L b s − = − − = +  ( ) ( ) ( ) ( ),1 1 f f s v f f b a a b L b b a − −  +   + +  (28) (Where ( ) ( )b gM a represents generalized Laguerrl polynomials) which is the result of [25]. In equation (20), setting 2, 0G H D= = = = and using definition of Rice polynomials ( ) ( ), , ,fH s v    , we get ( ) ( ) ( )0 : ; ,1 , ;:1:3 , : ; 1 ; , ;: 0:2! f g g f g f g f uA F a b vBg     = − − − + + + +    − +   Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 59 https://internationalpubls.com ( ) ( ) ( ) ( ), ! , , 1 f f f f f f a b H u v a        =   +   (29) which is the result of [33]. Putting 0, 1D E H G= = = = in equation (20), using suitable arrangement of variables and parameters and then applying the definition of Gegenbaur’s polynomials ( )u fC a , we get ( ) ( ) ( )0 ; ; , 2 ; :1;2 ,1 :0;1! : ;u ; 2 f g g g f g u ff A F a b Bg  =  − + + −   =  − +     ( ) ( ) ( ) !a 2 2 f f u f f f f a b C u a   −      (30) Also, putting 0, 1G E D H= = = = = in given equation (20) and applying the famous definitions of Shively’s Pseudo-Laguerre polynomials ( ),fR u  , we get ( ) ( ) ( )0 : ;:1;1 , : ; ;:0;1! f g g f g f gA F a b f gBg  = − − − +    − +   ( ) ( ) ( ) ( ) 2 ! , f f f f f f f a v b R v v a     =     (31) Putting 0, 2D G E H= = = = in equation (20), arranging variables and parameters and then applying the definition of Batemans’ polynomials ( ) ( ),u v fJ  , we get ( ) ( ) ( ) 2 2 0 : ; ; :1;1 , :0;2! : ; 1, 1; 2 f g g g f gf A F a bu Bg u v  =  − + −    − + + +     ( ) ( ) ( ) ( ) 2 , ! 1 1 2 1 2 f u f v v fu f u f u u a b J ub a u f   +    +  + +    =       + + +    (32) Putting 1D E G H= = = = in equation (20), arranging parameters suitably and applying the definition of Rainville’s polynomials ( ), ,f u v a , we obtain ( ) ( ) ( )0 1 1 : , ; , ;:2;2 ,2 2 2 2 :1;1! ; ; f g g u u f g f gA F a b Bg V V  =  − − + − + −  =       ( ) ( ) ( ) 1 , , f f f f f b a v u a    −       (33) Conclusion Hypergeometric series in one and more variables occur frequently in a wide variety of problems in theoretical physics, applied mathematics, engineering sciences, statistics and operation research.In our research note, we obtained a finite summation of triple hypergeometric series in terms of Kampe de Feriet’s double hypergeometric series. A number of finite sums of Kampe de Feriets’ double polynomials of higher order are obtained. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 60 https://internationalpubls.com Refrences [1] P. Appell, et al; Functions hypergèomètriques et hypersphèriues, Polynômes d’ Hermite, Gauthier- Villars, Paris, 1926. [2] G. Lauricella, Sulle funziont ipergeometriche a piu variabli, Rendiconti Circ. Mat. Palermo, 7, 111-158, 1893. [3] M. Turaev, Decomposition formula for Srivastava’s hypergemetric function AH on Saran functions, Journal of Computational and Applied Mathematics, 233, 842-846, 2009. [4] S. I. Bezrodnykh; Horn’s hypergeometric functions with three variables, Integral transforms and special functions, 2020, https://doi.org/10.1080/10652469.2020.1814770. [5] S.P. Chabra, and K.C. Rusia, A transformation formula for a general hypergeometric function of three variables, Jñanäbha, 9(10), 1980, 155-159. [6] V.L. Deshpande, Certain formulas associated with hypergeometric function of three variables, Pure and Applied Mathematica Sciences, 14, 1981, 39-45. [7] J. Choi, R. K. Parmar, Generalized Srivastava’s triple hypergeometric functions and their associated properties, Journal of Nonlinear Sciences and Applications, 10, 2017, 817-827. [8] M. A. Chaudhary, S. M. Zubair, On a class of incomplete gamma functions with applications, Chapman and Hall/CRC, Boca Raton, F1, 2002, 1. [9] J. S. Choi, R. K. Parmar, T. K. Pogány, Mathieu-type series built by (p, q)-extended Gaussian hypergeometric function, ArXiv, 2016, 9 pages. [10] J. S. Choi, R. K. Parmar, T. K. Pagany, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36, 2014, 357-385. [11] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (Eds.), NIST handbook of Mahtematical functions [with] CD-ROM [Windows, Macintosh and UNIX], US Department of Commerce, National Institute of Standards and Technology, Washinton, DC, (2010), Cambridge University Press, Cambridge, London and New York, 2010, 3] [12] R. K. Parmar, T. K. Pagàny, Extended Srivastava’s triple Hypergeometric HA,P,q function and relating bounding inequalities, J. Contemp. Math. Anal., 2016, In press), 1. [13] H. M. Srivastava, Hypergeometric functions of three variables, Ganita, 15, 1964, 97-108. [14] H. M. Srivastava, On transformations of certain hypergeometric functions of three variables, Publ. Math. Debrecen, 12, 1965, 65-74. [15] L. J. Slater, Generalized Hypergeometric functions, Cabridge University Press, Cambridge, 1966, 1. [16] R. Srivastava, Some generalizations of Pochhammer’s symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inf. Sci. 7, 2013, 2195-2206, 1. [17] R. Srivastava, Some classes of generating functions associated with a certain family of extended and generalized hypergeometric functions, Appl. Math. Comput. 243, 2014, 132-137. [18] H. M. Srivastava, A Çetinkaya et al, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput., 226, 2014, 484-491, 1, 1, 1. [19] H. M. Srivastava, R. K. Parmer, P, Chopra, A class of extended fractional derivative opeators and associated generating relations involving hypergeometric functions, Anioms, 1, 2012, 238-258, 1. [20] H. M. Srivastava, R. K. Parmar, M. M. Joshi, Extended Lauricella and Appl functions and their associated properties, Adv. Stud. Contemp. Math., 25, 2015, 151-165. [21] M. Saigo, On a property of the Appel hypergeometric functions F1, math. Rep. Kyushu Univ. 12, 1980, 63-67. [22] M. Saigo, On properties of the Appel hypergeometric functions F2 and F3 and the generalized Gauss functinos 3F2, Bull. Central Res. Inst. Fukuoka. Univ. 66, 1983, 27-32. [23] M. Saigo, On properties of hypergeometric functions of three variables, FM and FG, Rendi. Del. Circolo Mathmatico Di Palermo Seril II, Tomo XXXVII, 1988, 449-467. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 61 https://internationalpubls.com [24] M.P. Poudel, H.V. Harsh, N.P. Pahari, and D. Panthi, 2023. Kummer’s theorems, popular solutions and connecting formulas on Hypergeometric function. Journal of Nepal Mathematical Society, 6(1), 2023, 48-56. [25] M. P. Poudel, N. Pahari, G. Basnet, and R. Poudel. "Connection Formulas on Kummer’s Solutions and their Extension on Hypergeometric Function." Nepal Journal of Mathematical Sciences 4(2), 2023, 83- 88 [26] A Saboor, G. Rahman et al, A new extension of Srivastava’s triple hypergeometric functions and their associated properties, Analysis, 41 (1), 2021, 13-24. [27] H. L. Manocha and B. L. Sharma, Some formulae by means of fractional derivatives, Compositio Math., 18 (3), 1967, 229-234. [28] P. C. Munot, On Jacobi Polynomials, Proc. Camb. Philos. Soc., 65, 1969, 691-659. [29] M. I. Qureshi and M. A. Pathan, A none on hypergeometric polynomials J. Austral. Math. Soc. Ser. B, 26, 1984, 176-182. [30] M. A. Pathan, On a general triple hypergeometric series, Proc. Nat. Acad. Sci. India, A 47, 1977, 58- 60. [31] M. A. Pathan, On some transformation of triple hypergeometric series ( )3 F , Indian J Pure Appl. Math., 2 (4), 1978, 371-376. [32] Y. L. Luke, The special functions and their approximations-1, Academic Press, New York and London, 1969. [33] H. M. Srivastava, Certain formulas associated with generated Rice Polynomials-II, Annal. Polon. Math, 27, 1972, 73-83. [34] H. M. Srivastava, Certain formulas involving Appell functions, Comment Math. Univ. St. Paull, 21 (1), 1972, 73-99. [35] M.A. Rakha, and A.K. Rathie, Generalizations of Classical Summation Theorems for the Series 2F1 and 3F2 with Applications, Integral Transforms and Special Functions, 22,11,823-840,2011. [36] A. K. Thakur, S. K. Sahani and J. K. Kushwaha, Some Applications of Quadruple Hypergeometric Functions in function spaces, The Seybold Report, Vol. 17, No. 12,2022, 894-903,Doi: 10.5281/ zenodo. 7451049. [37] S.K,Sahani,et al.,Some Families of Bilinear and Bilateral Generating Functions in Function Space Associated with Hupergeometric Polynomials,Mathematicial Statistician and Engineering Applications, Vol.71,No.1,658-676,2022. [38] S.K.Sahani,et al,Recent Applications of Multiple Hypergeometric Transformations in Function Spaces and Associated Reduction Formulas, Tuijin Jishu/ Journal of Propulsion Technology,Vol.44,No.3,1522-1535,2023.