2 MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD Later on, in 977, the polynomials L n (x) were also studied by R. Panda [9]. It may be remarked here that in t

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1 SOOCHOW JOURNAL OF MATHEMATICS Volume 26, No., pp. 9-27, January 2 A STUDY OF BESSEL POLYNOMIALS SUGGESTED BY THE POLYNOMIALS L n (x) OF PRABHAKAR AND REKHA BY MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD Abstract. The present paper deals with a special case of Bessel polynomials suggested by the polynomials L n (x) ofparabhakar and Rekha. Certain integral representations, generating functions, Rodrigue's formula, recurrence relations, some characterizations and an orthogonality relation have been obtained for this polynomial.. Introduction The simple Bessel polynomial and the generalized one y n (x) = 2 F [;n +n ; x ] (.) 2 y n (a b x) = 2 F [;n a ; +n ; x ] (.2) b were investigated by Krall and Frink [8] in 949. Later on, in 965, Chatterjea [6] generalized (.2) and obtained certain generating functions for his generalized polynomial dened by where k is a positive integer. 2F [;n c + kn x] (.3) In 972, Prabhakar and Rekha [, ] considered a general class of polynomials suggested by Laguerre polynomials as dened below: L + +) n (x) =;(n n! Received July 3, 998 revised May 28, nx k= (;n) k x k k!;(k + +) : (.4)

2 2 MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD Later on, in 977, the polynomials L n (x) were also studied by R. Panda [9]. It may be remarked here that in terms of the classical Laguerre polynomials L n (x) =L() n (x): (.5) Motivated by the above mentioned works and those of Al-Salam [2] and Carlitz [5], we, in this paper, propose to study a special case of generalized Bessel polynomial (.2) suggested by (.3) and (.4). These polynomials turn out to be a special case of L n (x) also. 2. The Polynomial (y n (2 +( ; )n + 2 x)) The polynomial y n (2 + ( ; )n + 2 x) is a special case of (.2) and is dened as where n = 2 :::. y n (2 + ( ; )n + 2 x)= 2 F [;n n + + ; x ] (2.) 2 The polynomials (2.) are also special case of L n (x). In fact, by comparing the denitions (.4) and (2.), one easily gets y n (2 + ( ; )n + 2 x)=(n + +) n ( x 2 )n F [;n ;( +)n ; 2 x ] so that, obviously, y n (2 + ( ; )n + 2 x)=n(; x 2 )n L ;(+)n;; n ( 2 ): (2.2) x For = k, a + ve integer and = c ;, (2.) reduces to Chatterjea's generalization (.3) of Bessel polynomials. (.2). For =, = a;2 andx replaced by 2x, (2.) becomes Bessel's polynomial b For = and =, (2.) becomes the polynomial Y () n Al-Salam [2]. For =and =, (2.) becomes simple Bessel polynomial (.). considered by Sometimes we shall nd it convenient to consider the following polynomial: ( ) n (x) =x n y n (2 + ( ; )n + 2 ): (2.3) x

3 A STUDY OF BESSEL POLYNOMIALS SUGGESTED 2 For =, =, (2.3) reduces to () n (x) =xn Y () n ( ) (2.4) x a polynomial studied by Burchnall [3], Al-Salam [2] and Carlitz [4]. Notable contributions on Bessel polynomials are also from Carlitz [5] and Rainville [2]. For various notations and denitions one is referred to the book of Rainville [3] and the monumental works of Grosswald [7] and Srivastava and Mamocha [4]. 3. Integral Representations 2 x): It is easy to derive the following integral representations for y n (2+( ;)n+ t ; ( ; t) ; y n (2 + ( ; )n + 2 xt)dt " ;n n + + ;( + ) 3 F ; x + 2 = ;();() t ; ( ; t) ; y n (2 + ( ; )n + 2 x( ; t))dt " ;( + ) 3 F = ;();() ;n n + + ; x + 2 # # (3.) (3.2) e ;st t n+ ( + sxt 2 )n dt ;(n + +) = s n++ y n (2 + ( ; )n + 2 x) (3.3) x ;m;; ( ; x) ;n;; y m (2 + ( ; )m + 2 t x ) t y n (2 + ( ; )n + 2 ; x )dx sin (m + n + + );(m + n + + +) = ; sin (m + )sin(n + );(m + +);(n + ) y n+n (2 + ( ; )(m + n)+ + 2 t): (3.4) A two dimensional version of (3.4) is as follows: ZZ u+v u ;m;; v ;n;; ( ; u ; v) ;p;; y m (2 + ( ; )m + 2 t u )

4 22 MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD y n (2 + ( ; )n + 2 t v )y t p(2 + ( ; )p + 2 ; u ; v )dudv 2 sin ( + + );(m + n + p ) = sin sin sin ;(m + +);(n + +);(p + +) y m+n+p (2 + ( ; )(m + n + p) t) (3.5) where the integration is over the interior of the triangle bounded by u and v axes and the line u + v =. immediate. The extension of (3.5) to higher dimensions is 4. Generating Functions It is easy to derive the following generating functions: n= n= n= n= n= y n (2 ; n + 2 x) tn n! = et ( ; xt 2 );; (4.) y n (2 + 2 x) tn 2 =(; 2xt); 2 [ n! + p 2t ; 2xt ] e p + ;2xt (4.2) y n (2 + + ; n 2 x) tn n! = et ( ; xt 2 );;; (4.3) t n y n (2 ; n + 2 x) ; t 2 F " + xt 2( ; t) t n y n (2 + + ; n 2 x) ; t 2 F " + + An integral representation for (.2) due to Agarwal [] is y n (a b x) = ;(a + n ; ) # t a;2+n ( + xt b )n e ;t dt: (4.4) # xt : (4.5) 2( ; t) If we replace a by 2+( ; )n + and put b = 2 in the above integral representation, we obtain y n (2 + ( ; )n + 2 x)= ;(n + +) t n+ ( + xt 2 )n e ;t dt: (4.6) In formula (4.6) put n + = k, where n and k are integers. hence if we multiply through (;) k and then sum, we get k= (;) k y n (2 ; n + k 2 x)= + y x n(2 ; n 2 ): (4.7) +

5 A STUDY OF BESSEL POLYNOMIALS SUGGESTED 23 Similarly, wend k= (;) k y n (2 ; n + k 2 x)= k! In the same fashion one can derive the following formulae: k= k= k= (;) k 2k y n (2 ; n +2k 2 x)= (;) k 2k+ y n (3 ; n +2k 2 x)= t k k! y n(2 + ( ; )n + ; k 2 x) = ;(n + +) n= n= e ;t ( + xt 2 )n J (2 p t)dt: (4.8) e ;t ( + xt 2 )n cos tdt (4.9) e ;t ( + xt 2 )n sin tdt (4.) ( + xu 2 )n (u + t) n+ e ;u du (4.) t n n! y n(2 + ; 2n 2 x)=(+ xt 2 ) e 2t 2+xt (4.2) t n n! y n(2 + 2 x)=(; 2xt) ; 2 2 [ + p 2t ; 2xt ] e p + ;2xt : (4.3) 5. Rodrigues Formula Krall and Frink [8] gave the following Rodrigue's type formula for the Bessel polynomials y n (a b x): d n y n (a b x) =b ;n x 2;a e b x dx n (x2n+a;2 e ; b x ): (5.) Replacing a by 2+( ; )n + and putting b = 2 in (5.), the following Rodrigue's type nth derivative formula immediately follows: y n (2 + ( ; )n + 2 x)= 2 n x n;n+ e 2 x D n [x n+n+ ; 2 x ] D d dx : (5.2)

6 24 MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD 6. Recurrence Relations From (2.) it is easy to nd that y n (3 + ( ; )n + 2 x) ; y n (2 + ( ; )n + 2 x) = nx 2 y n;(3 + ( ; )(n ; ) x): (6.) This suggests the dierence formula y n (2 + ( ; )n + 2 x)= nx 2 y n;(3 + ( ; )(n ; ) x) (6.2) and k y n(2+(;)n+ 2 x)= n! (n ; x)! (x 2 )k y n;k (2+k+(;)(n;k)+k+ 2 x) where f() =f( +); f() and k+ f() = k f(). In particular (6.3) gives (6.3) n y n(2 + ( ; )n + 2 x)=n!( x 2 )n : (6.4) Now Newton's formula and (6.3) imply f( + ) = X r r r f() y n (2 + ( ; )n x) X n! = r r (n ; r)! ( x 2 )r y n;r (2 + r +( ; )(n ; r)+r + 2 x): (6.5) Also, from (2.) we nd that d dx y n( + ( ; )n + 2 x)= 2 n(n + +)y n;(3+( ; )(n ; ) x): (6.6) From (6.2) and (6.6), we nd that the polynomial given in (2.) satisfy the mixed equation y n (2 + ( ; )n + 2 x)= x d n + +dx y n(2 + ( ; )n + 2 x): (6.7)

7 A STUDY OF BESSEL POLYNOMIALS SUGGESTED 25 The following recurrence relation can easily be veried: y n+ (3 + ( ; )(n +)+ ; 2 x) ; y n (2 + ( ; )n + 2 x) = 2 x(n + n + +2)y n(3 + ( ; )n + ): (6.8) 7. Some Characterizations In this section we obtain some characterizations of the polynomial (2.) similar to those obtained by Al-Salam [2] for Bessel polynomial and are as given below: Theorem. Given a sequence ff n (2+( ; )n + 2 x)g of polynomials in x where deg. f n (2 + ( ; )n + 2 x)=n, and and are parameters such that d dx f n(2 + ( ; )n + 2 x)= 2 n(n + +)f n;(3+( ; )(n ; ) x) (7.) and f n (2 + ; )n + 2 ) =. Then f n (2 + ( ; )n + 2 x)=y n (2 + ( ; )n + 2 x): Another characterization is suggested by (6.2) as given in the following theorem: Theorem 2. Given a sequence of functions ff n (2 + ( ; )n + 2 x)g such that f n (2 + ( ; )n + 2 x)= 2 nxf n;(3 + ( ; )(n ; ) x) (7.2) f n (2 + ( ; )n + 2 ) = f (2 + 2 x)=: (7.3) Then f n (2 + ( ; )n + 2 x)=y n (2 + ( ; )n + 2 x): Next, equation (6.7) gives the following theorem:

8 26 MUMTAZ AHMAD KHAN AND KHURSHEED AHMAD Theorem 3. If the sequence ff n (2 + ( ; )n + 2 x)g, where f n (2+( ; )n + 2 x) is a polynomial of degree n in x, and are parameters, satises x d f n (2 + ( ; )n + 2 x)= n + +dx f n(2 + ( ; )n + 2 x) (7.4) such that f n (2 + ( ; )n 2 x)= 2 F [;n n + ; x ]: (7.5) 2 Then f n (2 + ( ; )n + 2 x)=y n (2 + ( ; )n + 2 x). Similarly (6.8) suggests yet another characterization of y n (2 + ( ; )n + 2 x) given in the form of the following theorem: that Theorem 4. Givenasequence of function ff n (2 + ( ; )n + 2 x)g such f n+ (3 + ( ; )(n +)+ ; 2 x) ; f n (2 + ( ; )n + 2 x) = 2 x(n + n + +2)f n(3 + ( ; )n + 2 x) (7.6) and f n (2 + 2 x)=for all x and. Then f n (2 + ( ; )n + 2 x)=y n (2 + ( ; )n + 2 x) 8. Orthogonality fy n (2 + ( ; )n + 2 x)g is a quasi-denite orthogonal polynomial set and adopting the procedure of Krall and Frink [8] the following orthogonality relation holds: 2i Z c ( z)y m (2 + ( ; )n + 2 z)y n (2 + ( ; )n + 2 z)dz = 2(;)n+ n!;f( ; )n + +g f( +)n + +g;(n + +) mn (8.) where ( z) = k= ;f( ; )n + +g ;fk +( ; )n + +g (;2 z )k (8.2)

9 A STUDY OF BESSEL POLYNOMIALS SUGGESTED 27 and the integration is around the unit circle. References [] R. P. Agarwal, On Bessel polynomials, Canadian Journal of Mathematics, 6(954), [2] W. A. Al-Salam, The Bessel polynomials, Duke Math. Journal, 24(957), [3] J. L. Burchnall, The Bessel polynomials, Canadian Journal of Mathematics, 3(95), [4] L. Carlitz, On the Bessel polynomials, Duke Math. Journal, 24(957), [5] L Carlitz, A characterization of the Laugerre polynomials, Monatshefte fur Mathematik, 66(962), [6] S. K. Chatterjea, Some generating funcitons, Duke math. Journal, 32(965), [7] E. Grosswald, Bessel polynomials, Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, New York, 978. [8] H. L. Krall and O. Frink, A new class of orthogonal polynomials: the Bessel polynomials, Transaction of the American Mathematical Society, 65(949), -5. [9] R. Panda, Anoteoncertain results involving the polynomials L ( ) n (x), Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 63:8(977), [] T. R. Prabhakar and S. Rekha, On a general class of polynomials suggested by Laguerre polynomials, Math. Student, 4(972), [] T. R. Prabhakar and S. Rekha, Some results on the polynomials L n (x), Rocky Mountain J. Math., 8(978), [2] E. D. Rainville, Generating functions for Bessel and related polynomials, Canadian Journal of Mathematics, 5(953), [3] E. D. Rainville, Special Functions, Macmillan, New York, Reprinted by Chelsea Publ. Co., Bronx, New York, 97. [4] H. M. Srivastava and H. L Manocha, A treatise on generating functions, John Wiley and Sons (Halsted Press), New York, Ellis Horwood, Chichester, 984. A/B-9, Medical Colony, Near Hadi Hasan Hall, A.M.U., Aligarh-222, U.P., India. Department of Applied Mathematics, Faculty of Engineering, A.M.U., Aligarh-222, U.P., India.

ON CERTAIN CHARACTERIZATIONS AND INTEGRAL REPRESENTATIONS OF CHATTERJEA S GENERALIZED BESSEL POLYNOMIAL

GLASNIK MATEMATIČKI Vol. 37(57)(22), 93 1 ON CERTAIN CHARACTERIZATIONS AND INTEGRAL REPRESENTATIONS OF CHATTERJEA S GENERALIZED BESSEL POLYNOMIAL Mumtaz Ahmad Khan and Khursheed Ahmad Aligarh Muslim University,