2 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
6 %%BoundingBox: 0 0 596 842
7 %%DocumentPaperSizes: a4
9 %DVIPSCommandLine: dvips -Pljfour -o laguerre.ps laguerre.dvi
10 %DVIPSParameters: dpi=600, compressed, comments removed
11 %DVIPSSource: TeX output 1997.02.16:1408
12 %%BeginProcSet: texc.pro
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645 AA2B>I<B9FCA23002809B31>I E end
648 %%Feature: *Resolution 600dpi
657 1 0 bop 404 573 a Fm(The)33 b(Laguerre)g(p)s(olynomials)c
658 Fl(L)1620 588 y Fk(n)1668 573 y Fm(\()p Fl(x)p Fm(\))k(are)f(de\014ned)
659 i(through)1286 851 y Fl(L)1352 866 y Fk(n)1400 851 y
660 Fm(\()p Fl(x)p Fm(\))28 b(=)f Fl(e)1707 809 y Fk(x)1773
661 851 y Fj(\001)1823 704 y Fi( )1926 783 y Fl(d)p 1899
662 827 107 4 v 1899 919 a(dx)2015 704 y Fi(!)2081 727 y
663 Fk(n)2144 851 y Fm(\()p Fl(x)2237 809 y Fk(n)2285 851
664 y Fl(e)2330 809 y Fh(\000)p Fk(x)2428 851 y Fm(\))257
665 1128 y Fg(Theorem:)404 1348 y Fl(L)470 1363 y Fk(n)517
666 1348 y Fm(\()p Fl(x)p Fm(\))33 b(satis\014es)g(the)g(recurrence)i
667 (relation)1668 1586 y Fl(L)1734 1601 y Ff(0)1774 1586
668 y Fm(\()p Fl(x)p Fm(\))28 b(=)f(1)907 1917 y Fl(L)973
669 1932 y Fk(n)p Ff(+1)1111 1917 y Fm(\()p Fl(x)p Fm(\))h(=)f(\(2)p
670 Fl(n)22 b Fm(+)h(1)e Fj(\000)i Fl(x)p Fm(\))g Fj(\001)e
671 Fl(L)2040 1932 y Fk(n)2088 1917 y Fm(\()p Fl(x)p Fm(\))h
672 Fj(\000)h Fl(n)2399 1876 y Ff(2)2461 1917 y Fj(\001)e
673 Fl(L)2576 1932 y Fk(n)p Fh(\000)p Ff(1)2714 1917 y Fm(\()p
674 Fl(x)p Fm(\))257 2130 y(for)k Fl(n)j Fj(\025)g Fm(0)e(and)g(the)g
675 (di\013eren)m(tial)d(equation)j Fl(x)8 b Fj(\001)g Fl(L)2046
676 2070 y Fe(00)2046 2155 y Fk(n)2093 2130 y Fm(\()p Fl(x)p
677 Fm(\))g(+)g(\(1)g Fj(\000)g Fl(x)p Fm(\))g Fj(\001)g
678 Fl(L)2699 2070 y Fe(0)2699 2155 y Fk(n)2746 2130 y Fm(\()p
679 Fl(x)p Fm(\))g(+)g Fl(n)g Fj(\001)g Fl(L)3137 2145 y
680 Fk(n)3184 2130 y Fm(\()p Fl(x)p Fm(\))28 b(=)f(0)257
681 2250 y(for)32 b(all)f Fl(n)d Fj(\025)g Fm(0.)257 2471
682 y Fg(Pro)s(of:)404 2691 y Fm(Let)41 b Fl(F)55 b Fm(:=)850
683 2625 y Fi(P)937 2651 y Fh(1)937 2716 y Fk(n)p Ff(=0)1101
684 2645 y Fk(L)1149 2653 y Fd(n)1191 2645 y Ff(\()p Fk(x)p
685 Ff(\))p 1101 2668 186 4 v 1162 2726 a Fk(n)p Ff(!)1324
686 2691 y Fj(\001)27 b Fl(z)1428 2655 y Fk(n)1517 2691 y
687 Fm(b)s(e)41 b(the)g(exp)s(onen)m(tial)f(generating)g(function)g(of)h
688 (the)257 2812 y(sequence)36 b(of)c(p)s(olynomials.)40
689 b(It)33 b(is)f(the)h(diagonal)d(series)j(of)f(the)h(p)s(o)m(w)m(er)h
690 (series)950 3084 y Fl(G)27 b Fm(:=)1257 2976 y Fh(1)1232
691 3001 y Fi(X)1185 3182 y Fk(m;n)p Ff(=0)1458 3016 y Fm(1)p
692 1426 3061 113 4 v 1426 3152 a Fl(m)p Fm(!)1571 3084 y
693 Fj(\001)22 b Fl(e)1666 3043 y Fk(x)1732 3084 y Fj(\001)1782
694 2938 y Fi( )1885 3016 y Fl(d)p 1857 3061 107 4 v 1857
695 3152 a(dx)1974 2938 y Fi(!)2039 2961 y Fk(m)2122 3084
696 y Fm(\()p Fl(x)2215 3043 y Fk(n)2263 3084 y Fl(e)2308
697 3043 y Fh(\000)p Fk(x)2407 3084 y Fm(\))g Fj(\001)g Fl(y)2569
698 3043 y Fk(m)2657 3084 y Fj(\001)f Fl(z)2755 3043 y Fk(n)257
699 3373 y Fm(Because)34 b(the)f(T)-8 b(a)m(ylor)31 b(series)i(dev)m
700 (elopmen)m(t)g(theorem)f(holds)f(in)h(formal)d(p)s(o)m(w)m(er)34
701 b(series)257 3494 y(rings)e(\(see)i([1],)f(section)f(2.16\),)g(w)m(e)i
702 (can)f(simplify)798 3771 y Fl(G)83 b Fm(=)g Fl(e)1162
703 3730 y Fk(x)1229 3771 y Fj(\001)1310 3664 y Fh(1)1285
704 3688 y Fi(X)1278 3870 y Fk(n)p Ff(=0)1428 3625 y Fi( )1535
705 3664 y Fh(1)1510 3688 y Fi(X)1494 3870 y Fk(m)p Ff(=0)1705
706 3704 y Fm(1)p 1673 3748 113 4 v 1673 3840 a Fl(m)p Fm(!)1817
707 3771 y Fj(\001)1867 3625 y Fi( )1971 3704 y Fl(d)p 1943
708 3748 107 4 v 1943 3840 a(dx)2059 3625 y Fi(!)2125 3648
709 y Fk(m)2208 3771 y Fm(\()p Fl(x)2301 3730 y Fk(n)2348
710 3771 y Fl(e)2393 3730 y Fh(\000)p Fk(x)2492 3771 y Fm(\))22
711 b Fj(\001)g Fl(y)2654 3730 y Fk(m)2720 3625 y Fi(!)2808
712 3771 y Fj(\001)f Fl(z)2906 3730 y Fk(n)958 4056 y Fm(=)83
713 b Fl(e)1162 4015 y Fk(x)1229 4056 y Fj(\001)1310 3948
714 y Fh(1)1285 3973 y Fi(X)1278 4154 y Fk(n)p Ff(=0)1411
715 4056 y Fm(\()p Fl(x)23 b Fm(+)f Fl(y)t Fm(\))1715 4015
716 y Fk(n)1761 4056 y Fl(e)1806 4015 y Fh(\000)p Ff(\()p
717 Fk(x)p Ff(+)p Fk(y)r Ff(\))2074 4056 y Fj(\001)g Fl(z)2173
718 4015 y Fk(n)958 4339 y Fm(=)1318 4272 y Fl(e)1363 4236
719 y Fh(\000)p Fk(y)p 1127 4316 523 4 v 1127 4408 a Fm(1)g
720 Fj(\000)h Fm(\()p Fl(x)f Fm(+)g Fl(y)t Fm(\))p Fl(z)257
721 4608 y Fm(W)-8 b(e)30 b(tak)m(e)f(o)m(v)m(er)h(the)g(terminology)c
722 (from)i(the)h(\\diag)p 2160 4608 30 4 v 34 w(rational")d(pap)s(er;)k
723 (here)g Fl(R)f Fm(=)e Fl(Q)p Fm([)p Fl(x)p Fm(])257 4729
724 y(and)h Fl(M)38 b Fm(=)28 b Fl(Q)p Fm([[)p Fl(x)p Fm(]])g(\(or,)h(if)d
725 (y)m(ou)i(lik)m(e)f(it)f(b)s(etter,)j Fl(M)39 b Fm(=)27
726 b Fl(H)8 b Fm(\()p Fl(C)f Fm(\),)28 b(the)g(algebra)e(of)h(functions)
727 257 4849 y(holomorphic)44 b(in)g(the)j(en)m(tire)e(complex)g(plane\).)
728 82 b Fl(G)50 b Fj(2)h Fl(M)10 b Fm([[)p Fl(y)t(;)17 b(z)t
729 Fm(]])46 b(is)f(not)h(rational;)257 4969 y(nev)m(ertheless)e(w)m(e)e
730 (can)f(pro)s(ceed)h(similarly)37 b(to)k(the)h(\\diag)p
731 2464 4969 V 33 w(series")g(pap)s(er.)69 b Fl(F)14 b Fm(\()p
732 Fl(z)3311 4933 y Ff(2)3351 4969 y Fm(\))41 b(is)257 5090
733 y(the)33 b(co)s(e\016cien)m(t)h(of)e Fl(t)1027 5054 y
734 Ff(0)1099 5090 y Fm(in)830 5335 y Fl(G)p Fm(\()p Fl(z)t(t;)1084
735 5268 y(z)p 1084 5312 50 4 v 1091 5404 a(t)1143 5335 y
736 Fm(\))c(=)1478 5268 y Fl(e)1523 5232 y Fh(\000)p Fk(z)s(t)p
737 1323 5312 477 4 v 1323 5404 a Fm(1)21 b Fj(\000)i Fl(z)1542
738 5375 y Ff(2)1604 5404 y Fj(\000)1714 5364 y Fk(xz)p 1714
739 5380 76 4 v 1739 5438 a(t)1837 5335 y Fj(2)28 b Fl(M)10
740 b Fm([[)p Fl(z)t(t;)2229 5268 y(z)p 2229 5312 50 4 v
741 2236 5404 a(t)2288 5335 y(;)17 b(z)t Fm(]])29 b(=)e Fl(M)10
742 b Fj(h)-17 b(h)p Fl(z)t(;)17 b(t)p Fj(i)-17 b(i)1852
745 2 1 bop 257 573 a Fm(The)34 b(denominator's)e(only)g(zero)h(is)f
746 Fl(t)c Fm(=)1830 534 y Fk(xz)p 1787 550 161 4 v 1787
747 607 a Ff(1)p Fh(\000)p Fk(z)1913 588 y Fc(2)1957 573
748 y Fm(.)44 b(W)-8 b(e)33 b(can)g(write)1048 829 y Fl(e)1093
749 788 y Fh(\000)p Fk(z)s(t)1241 829 y Fm(=)28 b Fl(e)1390
750 776 y Fh(\000)1477 749 y Fd(xz)1544 728 y Fc(2)p 1455
751 761 145 4 v 1455 811 a(1)p Fe(\000)p Fd(z)1565 797 y
752 Fc(2)1636 829 y Fm(+)1734 683 y Fi( )1799 829 y Fl(z)t(t)c
753 Fj(\000)2074 761 y Fl(xz)2178 725 y Ff(2)p 2016 806 260
754 4 v 2016 897 a Fm(1)e Fj(\000)h Fl(z)2236 868 y Ff(2)2286
755 683 y Fi(!)2374 829 y Fj(\001)e Fl(P)14 b Fm(\()p Fl(z)t(;)j(t)p
756 Fm(\))257 1088 y(with)47 b Fl(P)14 b Fm(\()p Fl(z)t(;)j(t)p
757 Fm(\))52 b Fj(2)g Fl(Q)p Fm([[)p Fl(z)t(t;)1240 1049
758 y Fk(xz)1316 1026 y Fc(2)p 1215 1065 161 4 v 1215 1123
759 a Ff(1)p Fh(\000)p Fk(z)1341 1104 y Fc(2)1385 1088 y
760 Fm(]])h Fj(\032)f Fl(Q)p Fm([[)p Fl(z)t(t;)17 b(x;)g(z)t
761 Fm(]])54 b(=)e Fl(M)10 b Fm([[)p Fl(z)t(t;)17 b(z)t Fm(]])54
762 b Fj(\032)f Fl(M)10 b Fj(h)-17 b(h)p Fl(z)t(;)17 b(t)p
763 Fj(i)-17 b(i)p Fm(.)87 b(This)257 1209 y(yields)33 b({)f(all)e
764 (computations)i(b)s(eing)g(done)h(in)f Fl(M)10 b Fj(h)-17
765 b(h)p Fl(z)t(;)17 b(t)p Fj(i)-17 b(i)33 b Fm({)534 1499
766 y Fl(G)p Fm(\()p Fl(z)t(t;)787 1432 y(z)p 787 1476 50
767 4 v 794 1567 a(t)847 1499 y Fm(\))83 b(=)1241 1432 y
768 Fl(e)1286 1379 y Fh(\000)1372 1352 y Fd(xz)1439 1331
769 y Fc(2)p 1351 1364 145 4 v 1351 1414 a(1)p Fe(\000)p
770 Fd(z)1461 1400 y Fc(2)p 1137 1476 477 4 v 1137 1567 a
771 Fm(1)22 b Fj(\000)g Fl(z)1356 1539 y Ff(2)1419 1567 y
772 Fj(\000)1528 1528 y Fk(xz)p 1528 1544 76 4 v 1553 1602
773 a(t)1646 1499 y Fm(+)1841 1432 y Fl(z)t(t)p 1753 1476
774 260 4 v 1753 1567 a Fm(1)g Fj(\000)h Fl(z)1973 1539 y
775 Ff(2)2045 1499 y Fj(\001)f Fl(P)14 b Fm(\()p Fl(z)t(;)j(t)p
776 Fm(\))968 1780 y(=)1242 1712 y(1)p 1137 1756 V 1137 1848
777 a(1)22 b Fj(\000)g Fl(z)1356 1819 y Ff(2)1428 1780 y
778 Fj(\001)g Fl(e)1523 1727 y Fh(\000)1610 1700 y Fd(xz)1677
779 1679 y Fc(2)p 1588 1712 145 4 v 1588 1762 a(1)p Fe(\000)p
780 Fd(z)1698 1748 y Fc(2)1769 1780 y Fj(\001)1845 1672 y
781 Fh(1)1820 1697 y Fi(X)1819 1879 y Fk(j)t Ff(=0)1958 1658
782 y Fi(\022)2131 1712 y Fl(x)p 2029 1756 260 4 v 2029 1848
783 a Fm(1)g Fj(\000)h Fl(z)2249 1819 y Ff(2)2309 1712 y
784 Fl(z)p 2309 1756 50 4 v 2316 1848 a(t)2369 1658 y Fi(\023)2430
785 1681 y Fk(j)2488 1780 y Fm(+)2684 1712 y Fl(z)t(t)p 2596
786 1756 260 4 v 2596 1848 a Fm(1)f Fj(\000)h Fl(z)2816 1819
787 y Ff(2)2888 1780 y Fj(\001)f Fl(P)14 b Fm(\()p Fl(z)t(;)j(t)p
788 Fm(\))257 2044 y(Here,)34 b(the)f(co)s(e\016cien)m(t)g(of)f
789 Fl(t)1284 2008 y Ff(0)1356 2044 y Fm(is)1380 2262 y Fl(F)14
790 b Fm(\()p Fl(z)1544 2220 y Ff(2)1583 2262 y Fm(\))28
791 b(=)1868 2194 y(1)p 1763 2238 V 1763 2330 a(1)21 b Fj(\000)i
792 Fl(z)1982 2301 y Ff(2)2054 2262 y Fj(\001)f Fl(e)2149
793 2209 y Fh(\000)2236 2182 y Fd(xz)2303 2161 y Fc(2)p 2214
794 2194 145 4 v 2214 2244 a(1)p Fe(\000)p Fd(z)2324 2230
795 y Fc(2)257 2473 y Fm(hence)1436 2613 y Fl(F)14 b Fm(\()p
796 Fl(z)t Fm(\))28 b(=)1866 2545 y(1)p 1780 2589 221 4 v
797 1780 2681 a(1)22 b Fj(\000)g Fl(z)2032 2613 y Fj(\001)g
798 Fl(e)2127 2569 y Fh(\000)2214 2542 y Fd(xz)p 2192 2554
799 111 4 v 2192 2595 a Fc(1)p Fe(\000)p Fd(z)404 2818 y
800 Fm(It)33 b(follo)m(ws)f(that)h(\(1)22 b Fj(\000)h Fl(z)t
801 Fm(\))1339 2782 y Ff(2)1402 2818 y Fj(\001)1480 2779
802 y Fk(d)p 1462 2795 72 4 v 1462 2853 a(dz)1544 2818 y
803 Fl(F)36 b Fj(\000)23 b Fm(\(1)g Fj(\000)g Fl(x)g Fj(\000)g
804 Fl(z)t Fm(\))g Fj(\001)f Fl(F)43 b Fm(=)28 b(0.)45 b(This)34
805 b(is)f(equiv)-5 b(alen)m(t)33 b(to)257 2939 y(the)g(claimed)e
806 (recurrence.)404 3059 y(Starting)e(from)h(the)h(closed)g(form)e(for)i
807 Fl(F)14 b Fm(,)30 b(w)m(e)i(compute)f(a)g(linear)e(relation)g(for)h
808 (the)257 3180 y(partial)h(deriv)-5 b(ativ)m(es)32 b(of)g
809 Fl(F)14 b Fm(.)44 b(W)-8 b(rite)31 b Fl(@)1637 3195 y
810 Fk(x)1710 3180 y Fm(=)1843 3140 y Fk(d)p 1823 3156 77
811 4 v 1823 3214 a(dx)1942 3180 y Fm(and)h(\001)2212 3195
812 y Fk(z)2280 3180 y Fm(=)c Fl(z)2461 3140 y Fk(d)p 2443
813 3156 72 4 v 2443 3214 a(dz)2525 3180 y Fm(.)44 b(One)33
814 b(computes)1674 3357 y Fl(F)41 b Fm(=)28 b(1)21 b Fj(\001)h
815 Fl(F)1403 3591 y Fm(\(1)f Fj(\000)i Fl(z)t Fm(\))g Fj(\001)f
816 Fl(@)1822 3606 y Fk(x)1866 3591 y Fl(F)42 b Fm(=)27 b
817 Fj(\000)p Fl(z)g Fj(\001)22 b Fl(F)1401 3850 y Fm(\(1)g
818 Fj(\000)h Fl(z)t Fm(\))1698 3802 y Ff(2)1759 3850 y Fj(\001)f
819 Fl(@)1865 3809 y Ff(2)1860 3875 y Fk(x)1905 3850 y Fl(F)42
820 b Fm(=)27 b Fl(z)2162 3809 y Ff(2)2225 3850 y Fj(\001)21
821 b Fl(F)1114 4110 y Fm(\(1)h Fj(\000)h Fl(z)t Fm(\))1411
822 4061 y Ff(2)1472 4110 y Fj(\001)f Fm(\001)1603 4125 y
823 Fk(z)1643 4110 y Fl(F)41 b Fm(=)28 b(\(\(1)22 b Fj(\000)g
824 Fl(x)p Fm(\))p Fl(z)28 b Fj(\000)22 b Fl(z)2411 4068
825 y Ff(2)2452 4110 y Fm(\))g Fj(\001)f Fl(F)1047 4369 y
826 Fm(\(1)h Fj(\000)h Fl(z)t Fm(\))1344 4320 y Ff(3)1405
827 4369 y Fj(\001)f Fl(@)1506 4384 y Fk(x)1550 4369 y Fm(\001)1631
828 4384 y Fk(z)1671 4369 y Fl(F)42 b Fm(=)27 b(\()p Fj(\000)p
829 Fl(z)h Fm(+)22 b Fl(xz)2269 4328 y Ff(2)2331 4369 y Fm(+)g
830 Fl(z)2478 4328 y Ff(3)2519 4369 y Fm(\))g Fj(\001)f Fl(F)257
831 4546 y Fm(Solv)m(e)33 b(a)f(homogeneous)h(4)22 b Fj(\002)h
832 Fm(5)32 b(system)h(of)g(linear)d(equations)j(o)m(v)m(er)h
833 Fl(Q)p Fm(\()p Fl(x)p Fm(\))f(to)f(get)891 4736 y(\(1)22
834 b Fj(\000)g Fl(z)t Fm(\))1187 4687 y Ff(3)1248 4736 y
835 Fj(\001)1298 4639 y Fi(\020)1348 4736 y Fm(\(1)g Fj(\000)g
836 Fl(x)p Fm(\))h Fj(\001)f Fl(@)1773 4751 y Fk(x)1817 4736
837 y Fl(F)36 b Fm(+)22 b Fl(x)g Fj(\001)g Fl(@)2197 4695
838 y Ff(2)2192 4760 y Fk(x)2238 4736 y Fl(F)35 b Fm(+)22
839 b(\001)2515 4751 y Fk(z)2556 4736 y Fl(F)2633 4639 y
840 Fi(\021)2710 4736 y Fm(=)27 b(0)257 4913 y(Divide)32
841 b(b)m(y)h(the)g(\014rst)g(factor)f(to)g(get)1144 5090
842 y(\(1)22 b Fj(\000)h Fl(x)p Fm(\))f Fj(\001)g Fl(@)1569
843 5105 y Fk(x)1613 5090 y Fl(F)36 b Fm(+)22 b Fl(x)h Fj(\001)f
844 Fl(@)1994 5049 y Ff(2)1989 5115 y Fk(x)2034 5090 y Fl(F)36
845 b Fm(+)22 b(\001)2312 5105 y Fk(z)2352 5090 y Fl(F)41
846 b Fm(=)28 b(0)257 5268 y(This)c(is)e(equiv)-5 b(alen)m(t)23
847 b(to)g(the)g(claimed)f(equation)g Fl(x)s Fj(\001)s Fl(L)2173
848 5208 y Fe(00)2173 5292 y Fk(n)2220 5268 y Fm(\()p Fl(x)p
849 Fm(\))s(+)s(\(1)s Fj(\000)s Fl(x)p Fm(\))s Fj(\001)s
850 Fl(L)2796 5208 y Fe(0)2796 5292 y Fk(n)2843 5268 y Fm(\()p
851 Fl(x)p Fm(\))s(+)s Fl(n)s Fj(\001)s Fl(L)3214 5283 y
852 Fk(n)3260 5268 y Fm(\()p Fl(x)p Fm(\))28 b(=)257 5388
853 y(0.)1852 5637 y(2)p eop
855 3 2 bop 257 573 a Fb(References)62 792 y Fm(1)146 b(Bruno)46
856 b(Haible:)69 b(D-\014nite)45 b(p)s(o)m(w)m(er)i(series)g(in)e(sev)m
857 (eral)i(v)-5 b(ariables.)82 b Fa(Diploma)46 b(thesis,)501
858 912 y(University)35 b(of)g(Karlsruhe,)f(June)h(1989)p
859 Fm(.)43 b(Sections)33 b(2.15)f(and)h(2.22.)1852 5637
863 userdict /end-hook known{end-hook}if
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