BOLLETTINO UNIONE MATEMATICA ITALIANA

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1 BOLLETTINO UNIONE MATEMATICA ITALIANA S. K. Chatterjea An integral representation for the product of two generalized Bessel polynomials. Bollettino dell Unione Matematica Italiana, Serie 3, Vol. 18 (1963), n.4, p Zanichelli < L utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI

2 Bollettino dell Unione Matematica Italiana, Zanichelli, 1963.

3 An intégral représentation for the product of two generalized BesseJ polynomials Nota di S. K. CHATTERJEA (a Calcutta) (*) «s repie- Summary. The product of two generaltsed Bessel polynomials sented by a double intégral. 1. Introduction. A few years ago WATSON [1] gave an intégral représentation for the product L^(x)L\f\x)^ where L^\x) dénotes the gênerai LAGTJERRE polynomial of degree n. CARLITZ [2] has recently proved the following formula m l ' 2«+P+m+» r(q + w + i)n8 + n H- 1) n XJ ' 7T S r(a-t-p + în + w + l) Tt/' 7T/2 (1.1) / j e("*""")<?*+(a-p)6* 008»*+" <p C08«+P 6 * It/8 7T/2 * L -i«l So^ C0S 6 J ^ This formula is therefore a generalization of WATSON's considération. CARLITZ [2, p. 28] has also proved that T(a)r(6) Hc-t-d-t-l) riôît) nctwtï)* a; c + 1; " ) *^; ^ - ^ = (1.2) = j j **-»(l *)6-i e ic-<i;ei cosc+dô. o ir/2. *(a H- 6 ; c + d + 1; 2 cos Ô (atfeô» -H Î/(1 i)e- '))cz8 tf (*) Pervenuta alla Segretena dell'u. M. I. il 17 giugno 1963.

4 378 S K CHATTERJEA where 4>(a; c; x) îs defined by Hère we propose to give an intégral représentation for the product of two generalized BESSEL polynomials. In the terminology of KRALL and FRINK (3J the generalized BESSEL polynomial is l 1-3 ) y n fa «, b) = tf 9 {-n, a-l+n; ; xjb). In this connection we mention AL.-SAL.AM's resuit [4]: (1.4) Y< n *\u)y ( V(v) = oo = TXl*Ln) f*"»' W (l + «( -- -«g)) * 0 where 7 \x) = y n [x, a-h 2, 2) in the notation of KRALL,-FRINK. But the results of CARLITZ suggest that the product of two generalized BESSEL polynomials can be represented by a double intégral involving another polynomial of the same kind. 2. Polynomials related to the Bessel polynomials. We first consider the polynomials *Jc, x) related to the BESSEL polynomials y n (x, a, o), where * (Jl (c, x) is defined by [5]: (2-1 ) 4\.(c, *) = ^,F D (-n, c + w; _; x) To obtaintae KRALL-FRINK generalized BESSEL polynomial, we introduce the redundant parameter b by replacing x by ( xjb), put c = a l and multiply 0 w (a - 1, as/6) by m!/(a 1),. Now it follow from (2.1) that

5 AN INTEGRAL REPRESENTATION FOR THE PRODUCT, ETC. 379 Then we dérive *«i(c, x,o n (d, y) = v v (~ l) r+s &y s T(c -+- m -h r)t{d +- n -+- s) 1 ri si r(c)r(d) (m r)! (n - s)! * In other words, (2-3) mr[dfl> m (c, xyb n & y) = _ y v ( -1)'^'x'y'r(c-ntt+r)r(d+n+8){m+n-r-s)\ r(c+d+m+tt+r+s) ~~ r=q 8=o ri si r(c+d+w+w+r4 s) (m r)!(w s)! (ra+n-r s)\ Now we notice the results [6] : (2.4) and (25) r [x H- v H- 1) 2^+ 2(*+ v r r( t-+-ur(v + i) = 1 e(n- v) * COSH-+* de, ([A -h v > 1) TT/2 TT/2 FvSo =/^( 1 " ')-'*. (i*> 0, v> 0). 0 It follows therefore from (2.3) that (2.6) r(c)r(d}<6 M] (c, xyb n [d, y) = 1 TT/2 2*»+» r [ = /! ic+«i(l )d+«-i e (m-h)ôi COS'""*"" 6 0 7T/2 S 1)*... =-± ' 2 cos 8)-*. ; &^o fc! (m H- w fc)! * S ( k ) [xty t/(l -1) * e-<«>e«ded*, (c > 0, d ^ 0j.

6 380 S. K. CHATTERJEA In other words, r(c)r(di A i n/2 2m+*f r r (2.7) = j j to+m-i^ _ t)d+»-i e (m-«)ôt C0S»»+" 6. o jr/2 m + w ( l)*(c H- d) w+w+fe /s^e-et-t-^i- *) e e*\ k Finally using (2 2) we dérive r {c)rtd) ^, 2*«+n r /*, <2.8) == j j tc+fn~i(\ ^)d+«-i e (m-«)6* cos m +» 6. 0 TT/2, /. xte-fo H- «(1 ^) e et\ «W«(c + d t ^ J-)ded*. (c>0, d>0). 3. Formula for the product yjx, a, o). t/jt/, a', 6). In this section we express the formula (2.8), which is our main resuit, in terms of the generalized BESSEL polynomials. Indeed we dérive from (2 8) r(a-l)r(a'-l) ml ^ ni ml m\n\ ni 2 m +«+" / / : (a - l) w (a- - 1) J J ta+m -K ~ *) ' + M - 2 e(«-)6. cos«+» 6, 0 TT/2 * m f (a -f- o' 2, J?te-e. H-, (1 - *)e<>' j /2 cos ô) dôd*.

7 AN INTEGRAL REPRESENTATION FOR THE PRODUCT, ETC 381 I Now using (1.3) and (2.1) we easily obtain from the just now deduced from (2 8): resuit r(a -+- m l)r(a -+- n 1) r(a + a' + t W + M -2 *»l x > a > h^^ a '> b^ = 2m4n i r/2 to *v mlnl f f (3.1) = / w+n), / 1 < a + m -2(l *)a'+«-2 e (m-«)6* COS»»+«0. 0 TT/2 ib'txer-te -+- 6w(l )e * \ 2/*»+«^ g-^, a -+- a' - 1, 66'j dôd*. Next we know that KRALL-FRINK simple BESSEL polynomial y u (x) is obtained by putting a = b = 2. Thus we notice that y n (x, 2, 2) = y n (x). So using a = 6 = a'= 6'= 2, we dérive from (3.1)- 2 m + n (3.2) yjx)y {y) = -^ (m -+ M -+-1). 0 -TT/2 C 0 S ' From (3.2) we thus observe that the product of two simple BESSEL polynomials is represented by a double intégral involving a generalized BESSEL polynomial whose first parameter «a» is 3, while the redundant parameter «6» remains the same. REFERENCES [1] GL N WATSON, A note on the polynomials of Hermite and Laguerre, «J. Lond. Math. Soc», Vol. 13 (1938), pp [2] L. CARLITZ, An tntegral for the product of two Laguerre polynomials, *Boll. Un. Mat. ItaL», (8) Vol. 17 (1962), pp [3] H. L. KRALL and O. FRINK, A new class of orthogonal polynomials The Bessel polynomials, «Trans. Araei. Math. Soc», Vol. 65 (1949), pp [4] W.A. AL-SALAM, The Bessel polynomials, «Duke Math. Jour.» Vol. 24 (1957), pp [5] E.D. RAINVILLE, Spécial Functions, New York (i960), p [6] E.T. WHITTAKER and G. WATSON, A course of modem analysis, 4th. édition, Cambridge (1952), pp. 263, 253.