B OLLETTINO U NIONE M ATEMATICA I TALIANA Leonard Carlitz An integral for the product of two Laguerre polynomials. Bollettino dell’Unione Matematica Italiana, Serie 3, Vol. 17 (1962), n.1, p. 25–28. Zanichelli <http://www.bdim.eu/item?id=BUMI_1962_3_17_1_25_0> 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 http://www.bdim.eu/ Bollettino dell’Unione Matematica Italiana, Zanichelli, 1962. An intégral for the product of two Laguerre polynomials by LÉONARD CARLITZ (a Durham, TJ. S. A.) (*) Summary. - The product L^(x)L^(y) ts représentée by a double intégral. Let (1) Lm (x) = J dénote the LAGIJERRE polynomial of degree w. Then Lm (x)Ln (y) - rt<) gSfl ( - l)H- -fj (m-*-n — r — s) ! ~ F (m — r) ! (n — s) ! T(a -h r •+- 1)T^ -4- s -+- 1) ' Silice [4. p. 263] r(|jn-v -f- 1) 2^+v r — —— j e(^-v)9* it follows that Lm {x)Ln iy) = i — r(a -+- m -f- i)r(p -H w 7T 7T " 2 2" Cos ^"" ^ i!(w +—/c ç i)!r( c o g /c p (*) Pervemita alla Segreteria dellU.M.I. il 19 dicembre 1961. 26 LEONARD CARLTTZ Using (1) this reduces to (3>, , 2 + P + n (y) = -g K2 ,3) 7T 71 ¥ 2l r ( a + m + l)F(p + n r(<x .] ƒ ^ ivm+M y ) \ cos6 COS Cp In particular, when x = y (3) reduces to i w («} J^ (a:) - ^ ^ ^ (4) */ / 7T 7T " 2 ~"2 cos r(a+p + ( m — w ) ? C 0 R ( a — P) 0 cos*»-!^* cp c o s ^ + P 6 : c o s (6 — <p) c o s 6\ This formula may be compared with WATSOK'S formula [3]: Jbm (X) ±jn (X) F(a -+- m 21 IE 2 2 'o o As pointed out by WATSOIST. (5) includes BATLEY'S formula [2] 7T J (w+w) (6) Hm(x)Hn{x T fff" * [ TT (»H-n)! - J 1 Jtlm- sec <p) cos 1 2 0 cos ^ ( m AN INTEGRAL FOR THE PRODUCT OF TWO LAGUERRE POLYNOMIAXS 27 where It is not clear hovv to obtain a resalt like (6) from (8) or (4). However using (7) and (2) we get f w—2r-j-2s)ô* Since the formula ƒ• (m — 2r) ! {n — 2s) ! 7T 2 is valid for ail r, s pro\dded that 2r -h 2s <. m-*-n, extend the range o£ r, s and get _m\n\ we may r 1 Therefore TT . __ . , l ml ni f (8) B4*)fl^) - - ^ - ^ } 1 J •where xny% «W, 28 LEONARD C4RLITZ I n particular, when x = y, (8) reduces 1o Hm(x)H„(y) 2*lm+n) mlHj F i(»+n) 1 which is a disguised version of (6). In connection with (8) compaie [1, formula (5.5)]. We remark that for the confluent hypergeometric function <P(a; c ; x ) = 2 -j r=o \a we can prove that T(a)T(b) r(< = —— / i 0 7T ~~ 2 5EEFBEENCES [1] W. A. AL-SALAN and L. CAHLITZ, Some finite summation formulas for the classical orthogonal polynomials, « Eendiconti di Matematiea & délie sue appJicazioni », series 5, vol. 16 (1957), pp. 74-95. [2] W. JNT. BAIJ.EY, ^ln intégral représentation for the product of two Hermite polycmials, .Journal of the London Mathematical Society »-, vol. 13 (1988), pp. 202-203. [3] Gr. 'N. WATSON, A note on the polnomiais of Heemite and Laguerre, « Journal of the London Mathematica! (Society % vol. 13 (1938), pp. 204-209. [4] E. T. WHITTAKER and G. N. WATSON, A course of modem analy$isr 4 th édition, Cambridge. 1927.