Interntionl Bulletin of Mtheticl Reserch Volue XX, Issue X, Deceber 2014 Pges 16-27, ISSN: XXXX-XXXX Note on Sequence of Functions involving the Product of E γ,k () Mehr Chnd Deprtent of Mthetics Fteh College for Woen, Rpur Phul Bthind-151001, Indi ehr.jllndhr@gil.co Abstrct A rerkbly lrge nuber of opertionl techniques hve drwn the ttention of severl reserchers in the study of sequences of functions nd polynoils. In this sequel, here, we i to introduce new sequence of functions involving product of the generlized Mittg-Leffler function by using opertionl techniques. Soe generting reltions nd finite sution forul of the sequence presented here re lso considered. 1 Introduction The ide of representing the processes of clculus, differentition, nd integrtion, s opertors, is clled n opertionl technique, which is lso known s n opertionl clculus. Mny opertionl techniques involve vrious specil functions hve found soe significnt pplictions in vrious sub-fields of pplicble theticl nlysis. Severl pplictions of opertionl techniques cn be found in the probles of nlysis, in prticulr differentil equtions re trnsfored into lgebric probles, usully the proble of solving polynoil equtions. Since lst four decdes, nuber of workers like Chk5], Gould nd Hopper 11], Chtterje8], Singh27], Srivstv nd Singh29], Mittl15, 16, 17], Chndl6, 7], Srivstv24], Joshi nd Prjpt14], Ptil nd Thkre18] nd Srivstv nd Singh28] hve de deep reserch of the properties, pplictions nd different etensions of the vrious opertionl techniques. The key eleent of the opertionl technique is to consider differentition s n opertor D d d cting on functions. Liner differentil equtions cn then be recst in the for of n opertor vlued function F(D) of the opertor D cting on n unknown function which equls known function. Solutions re then obtined by king the inverse opertor of F cting on the known function. Indeed, rerkbly lrge nuber of sequences of functions involving vriety of specil functions hve been developed by ny uthors (see, for eple, 28]; for very recent work, see lso 1, 2, 3, 22, 23, 26]). Here we i t presenting new sequence of functions involving product of the E γ,k by using opertionl techniques. Soe generting reltions nd finite sution forul re lso obtined. For our purpose, we begin by reclling soe known functions nd erlier works. In 1971, by Mittl 15] gve the Rodrigues forul for the generlized Lgurre polynoils defined by Received: Deceber 2014 Keywords: Specil function, generting reltions, Mittg-Leffler, Sequence of function, finite sution forul, sybolic representtion. AMS Subject Clssifiction: 33E10, 44A45.
Note on Sequence of Functions involving the Product of E γ,k () 17 where p k () is polynoil in of degree k. T (α) kn () 1 n! α ep (p k ()) D n α+n ep ( p k ()) ], (1.1) Mittl16] lso proved the following reltion for (1.1) given by where s is constnt nd T s (s + D). T ( 1) kn () 1 n! α n ep (p k ()) T n s α ep ( p k ())], (1.2) In this sequel, in 1979, Srivstv nd Singh 28] studied sequence of functions V n (α) (;, k, s) defined by V n (α) (;, k, s) α ep {p k ()} θ n α ep { p k ()}] (1.3) n! By using the opertor θ (s + D), where s is constnt, nd p k () is polynoil in of degree k. Here, new sequence of function n (;, k, s) : 1 n! α { } n (;, k, s) is introduced s follows: pkj () ] (T,s ) n α, (1.4) where T,s (s + D), D d d, nd s re constnts, β 0, k j is finite nd non-negtive integer, p kj () re polynoils in of degree k j, where j 1, 2,..., r nd (.) is generlized Mittg-Leffler function. For the ske of copleteness, we recll the E γ,k (.). In 1903, the Swedish theticin Gost Mittg-Leffler 19] introduced the function E α (z), defined s E α (z) z n Γ(αn + 1) (1.5) where z is cople vrible nd Γ(.) is G function, α 0. The Mittg-Leffler function is direct generliztion of the eponentil function to which it reduces for α 0. For 0 < α < 1 it interpoltes between the pure eponentil nd hypergeoetric function 1 1 z. Its iportnce is relized during the lst two decdes due to its involveent in the probles of physics, cheistry, biology, engineering nd pplied sciences. Mittg-Leffler function nturlly occurs s the solution of frctionl order differentil eqution or frctionl order integrl equtions. The generliztion of E α (z) ws studied by Win 31] in 1905 nd he defined the function s
18 Mehr Chnd E α,β (z) z n Γ(αn + β) (α, β C; R(α) > 0, R(β) > 0) (1.6) Which is known s Win s function or generlized Mittg-Leffler function s E α,1 (z) E α (z). The forer ws introduced by Mittg-Leffler19] in connection with his ethod of sution of soe divergent series. In his ppers 19, 20], he investigted certin properties of this function. The function defined by (1.6) first ppered in the work of Win 31]. The function (1.6) is studied, ong others, by Win 31], Agrwl 4], Hubert 12] nd Hubert nd Agrwl 13] nd others. The in properties of these functions re given in the book by Erdelyi et l. (10], Section 18.1) nd ore coprehensive nd detiled ccount of Mittg-Leffler functions re presented in Dzherbshyn (9], Chpter 2). In 1971, Prbhkr 21] introduced the function E γ α,β (z) in the for of E γ α,β (z) (γ) n z n Γ(αn + β) n!, (1.7) where α, β, γ C; R(α) > 0, R(β) > 0), R(γ) > 0 nd (λ) n denotes the filir Pochher sybol or the shifted fctoril, since (1) n n! (n N 0 ) (λ) n Γ(λ + n) Γ(λ) { 1 (n 0; λ C {0}) λ(λ + 1)...(λ + n 1) (n N; λ C) (1.8) Recently generliztion of Mittge-Leffler function E γ α,β (z) of (1.7) studied by Srivstv nd Toovski 30] is defined s follows: E γ,k α,β (z) (γ) Kn z n Γ(αn + β) n!, (1.9) where α, β, γ C; R(α) > 0, R(β) > 0), R(γ) > 0; R(K) > 0 which, in the specil cse when K q(q (0, 1) N) nd in{r(α), R(β)} 0 (1.10) ws considered erlier by Shukl nd Prjpti 26]. Soe generting reltions nd finite sution forul of clss of polynoils or sequences of functions ( hve been obtined by using the properties of the differentil opertors. The opertors T,s (s + D) D d ) d is bsed on the work of Mittl 17], Ptil nd Thkre 18], Srivstv nd Singh 28].
Note on Sequence of Functions involving the Product of E γ,k () 19 Soe useful opertionl techniques re given below: ( ) ep (tt,s ) β f () β (1 β+s t) ) f ( (1 t) 1/), (1.11) ep (tt,s ) ( α n f () ) α (1 + t) 1+( ) f ( (1 + t) 1/), (1.12) (T,s ) n (uv) ( ) n (T,s ) n (v) ( T,1 ) (u), (1.13) nd (1 + D) (1 + + D) (1 + 2 + D)... (1 + ( 1) + D) β 1 ( β ) β 1 (1.14) (1 t) α β (1 t) ( ) α β (t). (1.15)! 2 Generting Reltions First generting reltion: (1 t) Second generting reltion: (1 + t) n (;, k, s) n t n (2.1) pkj () ] r ( p kj (1 t) 1/)] V (δ j,β j,γ j,k j,α n) n (;, k, s) n t n (2.2) 1+( pkj () ] r p kj ( (1 + t) 1/)]
20 Mehr Chnd Third generting reltion: (1 t) ( ) + n n (;, k, s) n t n (2.3) pkj () ] ( V p kj (1 t) 1/)] ( ) (,γ j,k j,α) n (1 t) 1/ ;, k, s Proof of the first generting reltion. We strt fro (1.4) nd consider n (;, k, s) t n α pkj () ] ep(tt,s ) α (2.4) Using the opertionl technique (1.11), Eqution (2.4) reduces to α (1 t) n (;, k, s) t n (2.5) pkj () ] α (1 t) pkj () ] r ( p kj (1 t) 1/)] p kj ((1 t) 1/)], which upon replcing t by t, yields (2.1). Proof of the second generting reltion. Agin fro (1.4), we hve n V (δ j,β j,γ j,k j,α n) n (;, k, s) t n α pkj () ] ep(tt,s ) α n.(2.6) Applying the opertionl technique (1.12), we get
Note on Sequence of Functions involving the Product of E γ,k () 21 α n V (δ j,β j,γ j,k j,α n) n (;, k, s) t n (2.7) (1 + t) pkj () ] α (1 + t) 1 1 pkj () ] r p kj ((1 + t) 1/)] (2.8) p kj ((1 + t) 1/)]. This proves (2.2). Proof of the third generting reltion. We cn write (1.4) s (T,s ) n α () ] n! α V (δ j, β j, γ j, K j, α) n (;, k, s) pkj () ] (2.9) or ep (t (T,s )),s (T ) n α,α n! ep (tt ) α V (δj,β j,γ j,k j n,α) (;, k, s) δ j,β pkj j () ] (2.10) t! (T,s ) +n α,s n! ep (tt ) α n (;, k, s) E γ j,k j δ j,β pkj j () ]. (2.11) Using the opertionl technique (1.11), Eqution (2.10) cn be written s:
22 Mehr Chnd t! which, upon using (2.8), gives (T,s ) +n n! α (1 t) α pkj () ] (2.12) ) ( n (1 t) 1/ ;, k, s ) ( p kj (1 t) 1/)] Therefore, we hve t ( + n)!!n! α V (δj,β j,γ j,k j,α) +n (;, k, s) () ] n α (1 t) ) ( (2.13) p kj (1 t) 1/)] (( )) (1 t) 1/ ;, k, s. ( ) + n n +n (;, k, s) t Which, upon replcing t by t, proves the result (2.3). (1 t) pkj () ] ( (2.14) p kj (1 t) 1/)] n ( ) (1 t) 1/ ;, k, s. 3 Finite Sution Foruls First finite sution forul. n (;, k, s) Second finite sution forul. n (;, k, s) Proof of the first finite sution forul. 1 (! ( ) α ) V (δ j,β j,γ j,k j,0) n (;, k, s). (3.1) 1! ( ) ( ) α β V (δ j,β j,γ j,k j,β) n (;, k, s). (3.2)
Note on Sequence of Functions involving the Product of E γ,k () 23 Fro Eqution (1.4), we hve n (;, k, s) 1 n! α pkj () ] (T,s ) n α 1. (3.3) Using the opertionl technique (1.13), we hve n (;, k, s) (3.4) 1 n! α δ j,β pkj j () ] ( ) n (T,s ) n ( ) T,1 ( α 1 ) 1 n! α pkj () ] n!! (n )! (n ) (s + D) (s + + D) (s + 2 + D)... (s + (n 1) + D)] pkj () ] (1 + D) (1 + + D) (1 + 2 + D)... (1 + ( 1) + D)] ( α 1). Using the result (1.14), we hve n (;, k, s) (3.5) δ j,β pkj j () ] n n 1 1 (n )!n (s + i + D)! ( α ). i0 Put α 0 nd replcing n by n in (3.3), we get V (δ j,β j,γ j,k j,0) n (;, k, s) 1 (n )! pkj () ] (T,s ) n. (3.6) This gives 1 (n )! (T,s ) n V (,γ,k,0) n (;, k, s) δ j,β pkj j () ]. (3.7)
24 Mehr Chnd n 1 1 (s + i + D) (n )! i0 Fro Equtions (3.5) nd (3.8), we hve the in result. (δj,βj,γj,kj,0) V n (;, k, s) ( n) pkj () ]. (3.8) Proof of the second finite sution forul. Eqution (1.4) cn be written s n (;, k, s) t n α Applying the (1.11) to the Eqution (3.9), we hve pkj () ] ( ep tt (,s) ) α. (3.9) α n (;, k, s) t n (3.10) pkj () ] α (1 t) α+ s ( p kj (1 t) 1/)] (1 t) pkj () ] r Using the result fro Eqution (1.15), Eqution (3.10) reduces to p kj ((1 t) 1/)]. (1 t) β+s ) ( ) α β n (;, k, s) t n (3.11) ( t)! pkj () ] r ( p kj (1 t) 1/)] ( ) α β ( t)! β pkj () ] ep (tt,s ) β
Note on Sequence of Functions involving the Product of E γ,k () 25 ( ) α β ( ) t n+!n! β pkj () ] (T,s ) n β n ( ) α β ( ) t n! (n )! β Now equting the coefficient of t n, we get pkj () ] (T,s ) n β. n ( ) α β n (;, k, s) (3.12) ( )! (n )! β δ j,β pkj j () ] (T,s ) n β. Using the Eqution (1.4) in (3.12), we hve the result (3.2). 4 Specil Cses (I) If we tke r 1, γ j K j 1, then the results estblished in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduce to the known results in 1]. (II) If we choose γ j K j δ j β j 1, the Mittge-Leffler function reduced to ep(z) i.e. E 1,1 1,1 ep(z), then the results in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduced to the new results involving (ep(z)) r. (III) If we choose γ j K j 1, δ j 2, β j 1, the Mittge-Leffler function reduced to cosh( z) i.e. E 1,1 2,1 cosh( z), then the results in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduced to the new results involving (cosh( z)) r. 5 Conclusion In this pper, we hve presented new sequence of functions involving the product of the E γ,k (.) by using opertionl techniques. With the help of our in sequence forul, soe generting reltions nd finite sution forul of the sequence re lso presented here. Our sequence forul is iportnt due to presence of E γ,k (.). On ccount of the ost generl nture of the Eγ,K (.) lrge nuber of sequences nd polynoils involving sipler functions cn be esily obtined s their specil cses but due to lck of spce we cn not ention here.
26 Mehr Chnd References 1] Agrwl, P. nd Chnd, M. (2013). Grphicl Interprettion of the New Sequence of Functions Involving Mittge-Leffler Function Using Mtlb, Aericn Journl of Mthetics nd Sttistics 2013, 3(2): 73-83 DOI: 10.5923/j.js.20130302.02. 2] Agrwl, P. nd Chnd, M. (2013), On new sequence of functions involving p F q, South Asin Journl of Mthetics, Vol. 3(3): 199 210. 3] Agrwl, P. nd Chnd, M. (ccepted 2013), A NEW SEQUENCE OF FUNCTIONS INVOLVING A PRODUCT OF THE p F q, Mtheticl Sciences And Applictions E-Notes Volue 1. 4] Agrwl, R.P., (1953). A propos dune note de M. Pierre Hubert, C.R. Acd. Sci. Pris, 236, 2031-2032. 5] Chk, A. M., (1956) A clss of polynoils nd generliztion of stirling nubers, Duke J. Mth., 23, 45-55. 6] Chndel, R.C.S., (1973) A new clss of polynoils, Indin J. Mth., 15(1), 41-49. 7] Chndel, R.C.S., (1974) A further note on the clss of polynoils Tn α,k 39-48. (, r, p), Indin J. Mth.,16(1), 8] Chtterje, S. K., (1964) On generliztion of Lguerre polynoils, Rend. Mt. Univ. Pdov, 34, 180-190. 9] Dzherbshyn, M.M., (1966). Integrl Trnsfors nd Representtions of Functions in the Cople Plne, Nuk, Moscow, (in Russin). 10] Erdelyi, A., Mgnus,W., Oberhettinger, F. nd Tricoi, F. G. (1955). Higher Trnscendentl Functions, Vol. 3, McGrw - Hill, New York, Toronto nd London. 11] Gould, H. W. nd Hopper, A. T., (1962) Opertionl foruls connected with two generliztions of Herite polynoils, Duck Mth. J., 29, 51-63. 12] Hubert, P., (1953). Quelques resultnts retifs l fonction de Mittg-Leffler, C.R. Acd. Sci. Pris, 236, 1467-1468. 13] Hubert, P. nd Agrwl, R.P., (1953). Sur l fonction de Mittg-Leffler et quelques unes de ses generliztions, Bull. Sci. Mth., (Ser.II), 77, 180-185. 14] Joshi, C. M. nd Prjpt, M. L., (1975) The opertor T,k, nd generliztion of certin clssicl polynoils, Kyungpook Mth. J., 15, 191-199. 15] Mittl, H. B., (1971) A generliztion of Lguerre polynoil, Publ. Mth. Debrecen, 18, 53-58. 16] Mittl, H. B., (1971), Opertionl representtions for the generlized Lguerre polynoil, Glsnik Mt.Ser III, 26(6), 45-53. 17] Mittl, H. B., (1977), Biliner nd Bilterl generting reltions, Aericn J. Mth., 99, 23-45. 18] Ptil, K. R. nd Thkre, N. K., (1975), Opertionl foruls for function defined by generlized Rodrigues forul-ii, Sci. J. Shivji Univ. 15, 1-10.
Note on Sequence of Functions involving the Product of E γ,k () 27 19] Mittg-Leffler, G.M., (1903). Une generlistion de lintegrle de Lplce-Abel, C.R. Acd. Sci. Pris (Ser. II), 137, 537-539. 20] Mittg-Leffler, G.M., (1905). Sur l representtion nlytiqie dune fonction onogene (cinquiee note), Act Mthetic, 29, 101-181. 21] Prbhkr, T.R. (1971). A Singulr integrl eqution with generlized Mittg-Leffler function in the kernel, Yokoh Mth. J., Vol. 19, pp. 715. 22] Prjpti, J.C. nd Ajudi, N.K., (2012), On sequence of functions nd their MATLAB ipleenttion, Interntionl Journl od Physicl, Cheicl nd Mtheticl Sciences, Vol. No. 2, ISSN:2278-683, p.p. 24-34. 23] Slebhi, I.A., Prjpti, J.C. nd Shukl, A.K., On sequence of functions, Coun. Koren Mth. 28(2013), No.1,p.p. 123-134. 24] Shrivstv, P. N., (1974), Soe opertionl foruls nd generlized generting function, The Mth. Eduction, 8, 19-22. 25] Srivstv,H. M. nd Choi,J., (2012), Zet nd q-zet Functions nd Associted Series nd Integrls, Elsevier Science Publishers, Asterd, London nd New York. 26] Shukl, A. K. nd Prjpti J. C., (2007) On soe properties of clss of Polynoils suggested by Mittl, Proyecciones J. Mth., 26(2), 145-156. 27] Singh, R. P., (1968), On generlized Truesdell polynoils, Rivist de Mthetic, 8, 345-353. 28] Srivstv, A. N. nd Singh, S. N., (1979) Soe generting reltions connected with function defined by Generlized Rodrigues forul, Indin J. Pure Appl. Mth., 10(10), 1312-1317. 29] Srivstv, H. M. nd Singh, J. P., (1971) A clss of polynoils defined by generlized, Rodrigues forul, Ann. Mt. Pur Appl., 90(4), 75-85. 30] Srivstv, H.M. nd Toovski, Z.(2009). Frctionl clculus with n integrl opertor contining generlized Mittg-Leffler function in the kernel. Appl. Mth. Coput. 211(1), 198-210. 31] Win, A., (1905). Über den Fundentl stz in der Theorie der Funcktionen, E α(), Act Mthetic, 29, 191-201.