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A Note on a Three Variables Analogue of Bessel Polynomials
 

A Note on a Three Variables Analogue of Bessel Polynomials

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The present paper deals with a study of a three variables analogue of Bessel polynomials. ...

The present paper deals with a study of a three variables analogue of Bessel polynomials.
Certain representations, a Schlafli’s contour integral, a fractional integral, Laplace transformations, some
generating functions and double and triple generating functions have been obtained.

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    A Note on a Three Variables Analogue of Bessel Polynomials A Note on a Three Variables Analogue of Bessel Polynomials Document Transcript

    • International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 1 | A Note on a Three Variables Analogue of Bessel Polynomials Bhagwat Swaroop Sharma I. Introduction In 1949 Krall and Frink [12] initiated a study of simple Bessel polynomial      2 x ;n;1,nF(x)Y o2n (1.1) and generalized Bessel polynomial        b x ;n;1a,nFxb,a,Y o2n (1.2) These polynomials were introduced by them in connection with the solution of the wave equation in spherical coordinates. They are the polynomial solutions of the differential equation. x2 y (x) + (ax + b) y (x) = n (n + a – 1) y (x) (1.3) where n is a positive integer and a and b are arbitrary parameters. These polynomials are orthogonal on the unit circle with respect to the weight function n 0n x 2 )1n( )( i2 1 ),x(               . (1.4) Several authors including Agarwal [1], Al-Salam [2], Brafman [3], Burchnall [4], Carlitz [5], Chatterjea [6], Dickinson [7], Eweida [9], Grosswald [10], Rainville [15] and Toscano [19] have contributed to the study of the Bessel polynomials. Recently in the year 2000, Khan and Ahmad [11] studied two variables analogue y)(x,Y ),( n  of the Bessel polynomials (x)Y )( n  defined by      2 x ;1;n,nF(x)Y o2 )( n (1.5) The aim of the present paper is to introduce a three variables analogue c)b,a,z;y,(x,Y ),,( n  of (1.2) and to obtain certain results involving the three variables Bessel polynomial c)b,a,z;y,(x,Y ),,( n  . II. The Polynomials c)b,a,z;y,(x,Y ),,( n  : The Bessel polynomial of three variables c)b,a,z;y,(x,Y ),,( n  is defined as follows: c)b,a,z;y,(x,Y ),,( n          rsj rsjjsr srn 0j rn 0s n 0r c z b y a x !j!s!r 1n1n1nn                            (2.1) For z = 0, a = b = 2, (2.1) reduces to the two variables analogue y)(x,Y ),( n  of Bessel polynomials (1.5) as given below : y)(x,Yc)2,2,y,0;(x,Y ),( n ),,( n   (2.2) Similarly z)(x,Yb,2)2,z;0,(x,Y ),( n ),,( n   (2.3) Abstract: The present paper deals with a study of a three variables analogue of Bessel polynomials. Certain representations, a Schlafli’s contour integral, a fractional integral, Laplace transformations, some generating functions and double and triple generating functions have been obtained.
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 2 | z)(y,Y2)2,a,z;y,(0,Y ),( n ),,( n   (2.4) Also for  = – n – 1, and b = c = 2 z)(y,Y2)2,a,z;y,(x,Y ),( n ),,1n( n   (2.5) Similarly, z)(x,Y2)b,2,z;y,(x,Y ),( n ),1n,( n   (2.6) y)(x,Yc)2,2,z;y,(x,Y ),( n )1n,,( n   (2.7) where y)(x,Y ),( n        2 y 2 x !s!r 1n1nn rs rssr rn 0s n 0r                    (2.8) Also, for y = z = 0, a = 2, (2.1) reduces to the Bessel polynomials (x)Y )( n  as given below: (x)Yc)b,2,0,0;(x,Y )( n ),,( n   (2.9) where (x)Y )( n  is defined by (1.5). Similarly (y)Yc)2,a,0;y,(0,Y )( n ),,( n   (2.10) (z)Yc)2,2,z;0,(0,Y )( n ),,( n   (2.11) Also, for  =  = – n – 1, a = 2, we have (x)Yc)b,2,z;y,(x,Y )( n )1n1,n,( n   (2.12) Similarly (y)Yc)2,a,z;y,(x,Y )( n )1n,1,n( n   (2.13) (z)Y2)b,a,z;y,(x,Y )( n )1,n1,n( n   (2.14) III. Integral Representations It is easy to show that the polynomial c)b,a,z;y,(x,Y ),,( n  has the following integral representations:       dwdvdue c zw b yv a xu 1wvu 1n1n1n 1 wvu n nnn 000            = c)b,a,z;y,(x,Y ),,( n  (3.1) For z = 0, a = b = 2, (3.1) reduces to     dvdue 2 yv 2 xu 1vu 1n1n 1 vu n nn 00            = y)(x,Y ),( n  (3.2) a result due to Khan and Ahmad [11]. For y = z = 0,  replaced by a – 2 and a replaced by b, (3.1) becomes     dte b xt 1t n1a 1 xb,a,Y t n n2a 0 n             (3.3) a result due to Agarwal [1].           dzdydxc,b,a;z,y,xYztzysyxrx ,, n 1n1n1nr 0 s 0 t 0               cb,a,t;s,r,Y 111 ntsr nn,n, n nnn 3nnn      (3.4)           dwdvduc,b,a;zwyvu,xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0  
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 3 |                               c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.5) where F(3) [ ] is in the form of a general triple hypergeometric series F(3) [x, y, z] (cf. Srivastava [18], p. 428).           dwdvduc,b,a;zwyvu),1(xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.6)           dwdvduc,b,a;zw),v1(yu,xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.7)           dwdvduc,b,a);w1(z,yvu,xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.8)           dwdvduc,b,a;zw),v1(y,)u1(xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.9)           dwdvduc,b,a);w1(z,yv,)u1(xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.10)           dwdvduc,b,a);w1(z),v1(y,xuYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.11)           dwdvduc,b,a);w1(z),v1(y,)u1(xYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;1,n;1,n;1,n:;;::n F(3) (3.12)           dwdvduc,b,a;zuv,yuw,xvwYw1wv1vu1u ,, n 1111111 0 1 0 1 0                                 c z , b y , a x ;;;:;;:: ;1n1;n1;n:;;::n F(3) (3.13)
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 4 |          dudvdwc,b,a);v1(zu),u1(yw,)w1(xvYw1wv1vu1u ,, n 111111 1 0 1 0 1 0                                  c z , b y , a x ;;;:;;:: ;,1,n;,1,n;,1,n:;;::n F(3) (3.14) Particular Cases: Some interesting particular cases of the above results are as follows : (i) Taking  =  + 1,  =  + 1,  =  + 1,  =  =  = n in (3.5), we obtain           dwdvduc,b,a;zwyvu,xYw1wv1vu1u ,, n 1n1n1n1 0 1 0 1 0               cb,a,z;y,x,Y 111 n nn,n, n nnn 3     (3.15) which is equivalent to (3.4) (ii) Taking  =  =  = n + 1,  = ,  = ,  =  in (3.5), we get           dwdvduzwyvu,xYw1wv1vu1u ,, n 1n1n1n 1 0 1 0 1 0              zy,x,Y !n 0,0,0 n 1n1n1n 3    (3.16) (iii) Replacing  by  + n + 1 – ,  by  + n + 1 – ,  by  + n + 1 – , and putting  = ,  =  and  =  in (3.5), we get           dwdvduc,b,a;zwyvu,xYw1wv1vu1u ,, n 1n1n1n1 0 1 0 1 0                        cb,a,z;y,x,Y 1n1n1n 1n1n1n ,, n     (3.17) Similar particular cases hold for (3.6), (3.7), (3.8), (3.9), (3.10), (3.11) and (3.12). (iv) For z = 0, a = b = 2, results (3.4), (3.5), (3.6), (3.7) and (3.9) become         dydxy,xYysyxrx , n 1n1nr 0 s 0             sr,Y 11 ntsr nn, n nn 2nnn      (3.18)         dvduyvu,xYv1vu1u , n 1111 1 0 1 0                            2 y , 2 x ;;: ;1,n;1,n:n F 2;2:1 1;1: (3.19)         dvduyvu),1(xYv1vu1u , n 1111 1 0 1 0                             2 y , 2 x ;;: ;1,n;1,n:n F 2;2:1 1;1: (3.20)         dvdu)v1(yu),1(xYv1vu1u , n 1111 1 0 1 0                             2 y , 2 x ;;: ;1,n;1,n:n F 2;2:1 1;1: (3.21)
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 5 | Results (3.18), (3.19), (3.20) and (3.21) are due to Khan and Ahmad [11]. Also, using the integral (see Erdelyi et al. [8], vol. I, p. 14),     dtetzzsini2 t1z)(0     (3.22) and the fact that     !j!s!r zyxn zyx1 rsj jsr srn 0j rn 0s n 0r n         (3.23) we can easily derive the following integral representations for    cb,a,;zy,x,Y ,, n  :       dwdvdu c zw b yv a xu 1ewvu n wvunnn)0()0()0(                          cb,a,;zy,x,Y1n1n1nsinsinsin1i8 ,, n n   (3.24)         3 1n n1n1n1sinsinsin1      dwdvducb,a,; w cz , v by , u ax Yewvu ,, n wvu1n1n1n 000         n zyx1  (3.25) IV. Schlafli’s Contour Integral It is easy to show that dwdvdu c zw b yv a xu 1ewvu n wvunnn )0()0()0(                          cb,a,;zy,x,Yn1n1n1sinsinsin1i8 ,, n n   (4.1) Proof of (4.1) : We have   dwdvdu c zw b yv a xu 1ewvu i2 1 n wvunnn )0()0()0( 3                     dwdvduewvu i2 1 c z b y a x !j!s!r n wvurnsnjn )0()0()0( 3 rsj jsr srn 0j rn 0s n 0r                                         rnsnjn!j!s!r c z b y a x n rsj jsrsrn 0j rn 0s n 0r                           using Hankel’s formula (see A. Eerdelyi et al. [8], 1.6 (2)).   dtte i2 1 z 1 zt )(0      (4.2) Finally (4.1) follows from (2.1) after using the result     zcosecz1z  (4.3) for z = 0, a = b = 2, (4.1) reduces to dvdu b yv a xu 1evu n wvunn)0()0(                   yx,Yn1n1sinsin4 , n   (4.4) which is due to Khan and Ahmad [11].
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 6 | V. Fractional Integrals Let L denote the linear space of (equivalent classes of) complex – valued functions f(x) which are Lebesgue – integrable on [0, ],  < . For f(x) L and complex number  with Rl  > 0, the Riemann – Liouville fractional integral of order  is defined as (see Prabhakar [13], p. 72)     dt(t)ftx 1 f(x)I 1x 0      for almost all x  [0, ] (5.1) Using the operator I , Prabhakar [14] obtained the following result for Rl  > 0 and Rl  > –1.         kx;Zx 1kn 1kn kx;ZxI nn     (5.2) where  kx;Zn  is Konhauser’s biorthozonal polynomial. Khan and Ahmad [11] defined a two variable analogue of (5.1) by means of the following relation :              dvduvu,fvyux 1 yx,fI 11y 0 x 0 ,      (5.3) and obtained the following result :     yx,YyxI , n nn,             yx,Y 1n1n 1n1nyx , n nn      (5.4) In an attempt to obtained a result analogous to (5.4) for the polynomial    cb,a,;zy,x,Y ,, n  we first seek a three variable analogue of (5.1). A three variable analogue of I may be defined as                  dwdvduwv,u,fwzvyux 1 zy,x,fI 111z 0 y 0 x 0 ,,      (5.5) Putting      cb,a,z;y,x,Yzyxzy,,xf ,, n nnn   in (5.5), we obtain     cb,a,z;y,x,YzyxI ,, n nnn,,                  cb,a,z;y,x,Y 1n1n1n 1n1n1nzyx ,, n nnn      (5.6) VI. Laplace Transform In the usual notation the Laplace transform is given by        0asRldt,tfes:tfL st 0     (6.1) where f  L (0, R) for every R > 0 and f(t) = 0(eat ), t  . Khan and Ahmad [11] introduced a two variable analogue of (6.1) by means of the relation:      dvduvu,fes,s:vu,fL vsus 00 21 21    (6.2) and established the following results :               21 21 , n 1n1n s,s: vs 2y , us 2x YvuL      n n 2 n 1 2 yx1 1n1nsinsin ss      (6.3) and                21 n 21nn s,s: 2 yvs 2 xus 1vuL
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 7 |        yx,Y ss 1n1n , n1n 2 1n 1     (6.4) In an attempt to obtain results analogous to (6.3) and (6.4) for    cb,a,z;y,x,Y ,, n  we define a three variable analogue of (6.1) as follows      dwdvduwv,u,fes,s,s:wv,u,fL wsvsus 000 321 321   (6.5) Now, we have               321 321 ,, n 1n1n1n s,s,s:cb,a,; ws cz , vs by , us ax YwvuL         n n 3 n 2 n 1 31n y)x(1 1n1n1nsinsinsin sss1      (6.6) Similarly, we obtain                321 n 321nnn s,s,s: c zws b yvs a xus 1wvuL          cb,a,z;y,x,Y sss 1n1n1n ,, n1n 3 1n 2 1n 1     (6.7) VII. Generating Functions It is easy to derive the following generating functions for    cb,a,z;y,x,Y ,, n  :     111 tn,n,n n n 0n c zt 1 b yt 1 a xt 1ecb,a,z;y,x,Y !n t                        (7.1)    cb,a,z;y,x,Y !n t n,n, n n 0n     a xt4 11 t2 e a xt4 11 c zt2 1 a xt4 11 b yt2 1 a xt4 11 2 a xt4 1 11 2 1                                                   (7.2)    cb,a,z;y,x,Y !n t n,,n n n 0n     b yt4 11 t2 e b yt4 11 c zt2 1 b yt4 11 a xt2 1 b yt4 11 2 b yt4 1 11 2 1                                                   (7.3)    cb,a,z;y,x,Y !n t ,n,n n n 0n    
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 8 | c zt4 11 t2 11 2 1 e c zt4 11 b yt2 1 c zt4 11 a xt2 1 c zt4 11 2 c zt4 1                                                    (7.4)    cb,a,z;y,x,Y !n t n,n,n2 n n 0n         11 xta at xtac azt 1 xtab ayt 1 a xt 1e                         (7.5)    cb,a,z;y,x,Y !n t n,n2,n n n 0n         11 ytb bt ytbc bzt 1 ytba bxt 1 b yt 1e                         (7.6)    cb,a,z;y,x,Y !n t n2,n,n n n 0n         11 ztc ct ztcb cyt 1 ztca cxt 1 c zt 1e                         (7.7)           1cb,a,z;y,x,Y 1 1 cb,a,z;y,x,Y n,, n nk,, n k 0k        (7.8)           c,1ba,z;y,x,Y 1 1 cb,a,z;y,x,Y ,n, n ,nk, n k 0k        (7.9)           c,b,1az;y,x,Y 1 1 cb,a,z;y,x,Y ,,n n ,,nk n k 0k        (7.10) Using (3.1), we can also derive the following results :      cb,a,z;y,x,Y !k nk,, n k 0k            dwdvduw2Je c zw b yv a xu 1vu 1n1n 1 0 wvu n nn 000             (7.11)      cb,a,z;y,x,Y !k ,nk, n k 0k            dwdvduv2Je c zw b yv a xu 1wu 1n1n 1 0 wvu n nn 000             (7.12)      cb,a,z;y,x,Y !k ,,nk n k 0k            dwdvduu2Je c zw b yv a xu 1wu 1n1n 1 0 wvu n nn 000             (7.13)
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 9 |      cb,a,z;y,x,Y1 n2k,, n 2kk 0k         dwdvduwosce c zw b yv a xu 1vu 1n1n 1 wvu n nn 000             (7.14)      cb,a,z;y,x,Y1 ,n2k, n k2k 0k         dwdvduvosce c zw b yv a xu 1wu 1n1n 1 wvu n nn 000             (7.15)      cb,a,z;y,x,Y1 ,,n2k n 2kk 0k         dwdvduuosce c zw b yv a xu 1wu 1n1n 1 wvu n nn 000             (7.16)      cb,a,z;y,x,Y1 n12k,, n 12kk 0k         dwdvduwsine c zw b yv a xu 1vu 1n1n 1 wvu n nn 000             (7.17)      cb,a,z;y,x,Y1 ,n12k, n 1k2k 0k         dwdvduvsine c zw b yv a xu 1wu 1n1n 1 wvu n nn 000             (7.18)      cb,a,z;y,x,Y1 ,,n12k n 12kk 0k         dwdvduusine c zw b yv a xu 1wu 1n1n 1 wvu n nn 000             (7.19) VIII. Double Generating Functions The following double generating functions for    cb,a,z;y,x,Y ,, n  can easily be derived by using (3.1)        cb,a,z;y,x,Y ,nk,nm n km 0k0m                c,1b,1a;zy,,xY 11 1 n,n, n     (8.1)        cb,a,z;y,x,Y nk,,nm n km 0k0m                    1cb,,1a;zy,,xY 11 1 n,n, n (8.2)
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 10 |        cb,a,z;y,x,Y nk,nm, n km 0k0m                    1c,1ba,;zy,,xY 11 1 nn,, n (8.3)        cb,a,z;y,x,Y !k!m ,nk,nm n km 0k0m              dwdvduv2Ju2Je c zw b yv a xu 1w 1n 1 00 wvu n n 000             (8.4)        cb,a,z;y,x,Y !k!m nk,,nm n km 0k0m              dwdvduw2Ju2Je c zw b yv a xu 1v 1n 1 00 wvu n n 000             (8.5)        cb,a,z;y,x,Y !k!m nk,nm, n km 0k0m              dwdvduw2Jv2Je c zw b yv a xu 1u 1n 1 00 wvu n n 000            (8.6)      cb,a,z;y,x,Y1 ,n2k,nm2 n 2k2mkm 0k0m         dwdvduvcosuosce c zw b yv a xu 1w 1n 1 wvu n n 000            (8.7)      cb,a,z;y,x,Y1 n2k,,nm2 n 2k2mkm 0k0m         dwdvduwcosuosce c zw b yv a xu 1v 1n 1 wvu n n 000             (8.8)      cb,a,z;y,x,Y1 n2k,nm2, n 2k2mkm 0k0m         dwdvduwcosvosce c zw b yv a xu 1u 1n 1 wvu n n 000             (8.9)      cb,a,z;y,x,Y1 ,n2k,n1m2 n 2k12mkm 0k0m         dwdvduvcosuinse c zw b yv a xu 1w 1n 1 wvu n n 000             (8.10)      cb,a,z;y,x,Y1 n2k,,n1m2 n 2k12mkm 0k0m      
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 11 |   dwdvduwcosuinse c zw b yv a xu 1v 1n 1 wvu n n 000            (8.11)      cb,a,z;y,x,Y1 n2k,n1m2, n 2k12mkm 0k0m         dwdvduwcosvinse c zw b yv a xu 1u 1n 1 wvu n n 000            (8.12)      cb,a,z;y,x,Y1 ,n12k,nm2 n 12k2mkm 0k0m         dwdvduvsinuosce c zw b yv a xu 1w 1n 1 wvu n n 000             (8.13)      cb,a,z;y,x,Y1 n12k,,nm2 n 12k2mkm 0k0m         dwdvduwsinuosce c zw b yv a xu 1v 1n 1 wvu n n 000             (8.14)      cb,a,z;y,x,Y1 n12k,nm2, n 12k2mkm 0k0m         dwdvduwsinvosce c zw b yv a xu 1u 1n 1 wvu n n 000             (8.15)      cb,a,z;y,x,Y1 ,n12k,n1m2 n 12k12mkm 0k0m         dwdvduvsinuinse c zw b yv a xu 1w 1n 1 wvu n n 000             (8.16)      cb,a,z;y,x,Y1 n12k,,n1m2 n 12k12mkm 0k0m         dwdvduwsinuinse c zw b yv a xu 1v 1n 1 wvu n n 000             (8.17)      cb,a,z;y,x,Y1 n12k,n1m2, n 12k12mkm 0k0m         dwdvduwsinvinse c zw b yv a xu 1u 1n 1 wvu n n 000             (8.18)
    • A Note on a Three Variables Analogue of Bessel Polynomials | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 3 | Mar. 2014 | 12 | IX. Triple Generating Functions The following triple generating functions can easily be obtained by using (3.1):          cb,a,z;y,x,Y nj,nk,nm n jkm 0j0k0m                         1c,1b,1az;y,x,Y 111 1 n,n,n n (9.1)          cb,a,z;y,x,Y !j!k!m nj,nk,nm n jkm 0j0k0m                dwdvduw2Jv2Ju2Je c zw b yv a xu 1 000 wvu n 000           (9.2)      cb,a,z;y,x,Y1 n2j,n2k,nm2 n 2j2k2mjkm 0j0k0m         dwdvduwoscvoscuosce c zw b yv a xu 1 wvu n 000           (9.3)      cb,a,z;y,x,Y1 n12j,n12k,n1m2 n 12j12k12mjkm 0j0k0m         dwdvduwinsvinsuinse c zw b yv a xu 1 wvu n 000           (9.4)      cb,a,z;y,x,Y1 n2j,n12k,n1m2 n 2j12k12mjkm 0j0k0m         dwdvduwoscvinsuinse c zw b yv a xu 1 wvu n 000           (9.5)      cb,a,z;y,x,Y1 n12j,n2k,nm2 n 12j2k2mjkm 0j0k0m         dwdvduwinsvoscuosce c zw b yv a xu 1 wvu n 000           (9.6) X. Bessel Polynomials Of M-Variables The Bessel polynomials of m-variables    m21m21 ,----,, n a,----,a,a;x,----,x,xY m21  can be defined as follows:    m21m21 ,----,, n a,----,a,a;x,----,x,xY m21                             i i m 0i i m 0i ji m 0i rrrrrrn 0r rrn 0r rn 0r n 0r a x !r 1nn m211m21 m 21 3 1 21 (10.1) All the results of this paper can be extended for this m-variable Bessel polynomials. The only hinderance in their study is the representation of results in hypergeometric functions of m-variables.
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