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 legendre.ps legendre.dvi
10 %DVIPSParameters: dpi=600, compressed, comments removed
11 %DVIPSSource: TeX output 1997.02.16:1425
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635 FCEA03F02E3F7EAA33>I E end
638 %%Feature: *Resolution 600dpi
647 1 0 bop 404 573 a Fk(The)33 b(Legendre)h(p)s(olynomials)29
648 b Fj(P)1628 588 y Fi(n)1675 573 y Fk(\()p Fj(x)p Fk(\))k(are)g
649 (de\014ned)h(through)1199 856 y Fj(P)1262 871 y Fi(n)1309
650 856 y Fk(\()p Fj(x)p Fk(\))28 b(=)1647 789 y(1)p 1581
651 833 182 4 v 1581 924 a(2)1630 896 y Fi(n)1677 924 y Fj(n)p
652 Fk(!)1794 856 y Fh(\001)1844 710 y Fg( )1948 789 y Fj(d)p
653 1920 833 107 4 v 1920 924 a(dx)2036 710 y Fg(!)2102 733
654 y Fi(n)2165 856 y Fk(\()p Fj(x)2258 815 y Ff(2)2320 856
655 y Fh(\000)23 b Fk(1\))2507 815 y Fi(n)257 1137 y Fk(\(F)-8
656 b(or)40 b(a)h(motiv)-5 b(ation)38 b(of)j(the)g(2)1412
657 1101 y Fi(n)1500 1137 y Fk(in)f(the)h(denominator,)h(lo)s(ok)e(at)g
658 Fj(P)2815 1152 y Fi(n)2862 1137 y Fk(\()p Fj(x)p Fk(\))h(mo)s(dulo)e
659 (an)257 1258 y(o)s(dd)30 b(prime)f Fj(p)p Fk(,)h(and)g(observ)m(e)i
660 (that)e Fj(P)1631 1273 y Fi(n)1678 1258 y Fk(\()p Fj(x)p
661 Fk(\))e Fh(\021)g Fj(P)2005 1273 y Fi(p)p Fe(\000)p Ff(1)p
662 Fe(\000)p Fi(n)2232 1258 y Fk(\()p Fj(x)p Fk(\))62 b(mo)s(d)48
663 b Fj(p)30 b Fk(for)g(0)d Fh(\024)h Fj(n)g Fh(\024)g Fj(p)17
664 b Fh(\000)g Fk(1.)257 1378 y(This)33 b(w)m(ouldn't)g(hold)f(if)f(the)i
665 (2)1407 1342 y Fi(n)1486 1378 y Fk(factor)f(in)g(the)h(denominator)e(w)
666 m(eren't)j(presen)m(t.\))257 1605 y Fd(Theorem:)404 1832
667 y Fj(P)467 1847 y Fi(n)514 1832 y Fk(\()p Fj(x)p Fk(\))e(satis\014es)i
668 (the)f(recurrence)h(relation)1669 2075 y Fj(P)1732 2090
669 y Ff(0)1772 2075 y Fk(\()p Fj(x)p Fk(\))28 b(=)f(1)806
670 2417 y(\()p Fj(n)22 b Fk(+)g(1\))g Fh(\001)g Fj(P)1244
671 2432 y Fi(n)p Ff(+1)1381 2417 y Fk(\()p Fj(x)p Fk(\))28
672 b(=)f(\(2)p Fj(n)22 b Fk(+)g(1\))p Fj(x)h Fh(\001)e Fj(P)2185
673 2432 y Fi(n)2232 2417 y Fk(\()p Fj(x)p Fk(\))i Fh(\000)f
674 Fj(n)h Fh(\001)f Fj(P)2679 2432 y Fi(n)p Fe(\000)p Ff(1)2816
675 2417 y Fk(\()p Fj(x)p Fk(\))257 2636 y(for)33 b Fj(n)d
676 Fh(\025)f Fk(0)k(and)h(the)g(di\013eren)m(tial)d(equation)i(\(1)23
677 b Fh(\000)g Fj(x)2195 2600 y Ff(2)2235 2636 y Fk(\))g
678 Fh(\001)f Fj(P)2423 2576 y Fc(00)2409 2661 y Fi(n)2467
679 2636 y Fk(\()p Fj(x)p Fk(\))h Fh(\000)g Fk(2)p Fj(x)g
680 Fh(\001)g Fj(P)2976 2576 y Fc(0)2962 2661 y Fi(n)3008
681 2636 y Fk(\()p Fj(x)p Fk(\))g(+)g(\()p Fj(n)3357 2600
682 y Ff(2)3419 2636 y Fk(+)257 2756 y Fj(n)p Fk(\))g Fh(\001)e
683 Fj(P)488 2771 y Fi(n)535 2756 y Fk(\()p Fj(x)p Fk(\))28
684 b(=)g(0)k(for)g(all)f Fj(n)d Fh(\025)g Fk(0.)257 2983
685 y Fd(Pro)s(of:)404 3210 y Fk(Let)38 b Fj(F)51 b Fk(:=)838
686 3144 y Fg(P)926 3170 y Fe(1)926 3235 y Fi(n)p Ff(=0)1079
687 3210 y Fj(P)1142 3225 y Fi(n)1189 3210 y Fk(\()p Fj(x)p
688 Fk(\))26 b Fh(\001)g Fj(z)1449 3174 y Fi(n)1535 3210
689 y Fk(b)s(e)38 b(the)h(generating)e(function)h(of)f(the)i(sequence)i(of)
690 257 3330 y(p)s(olynomials.)g(It)32 b(is)h(the)g(diagonal)d(series)j(of)
691 f(the)h(p)s(o)m(w)m(er)h(series)949 3608 y Fj(G)28 b
692 Fk(:=)1257 3500 y Fe(1)1232 3525 y Fg(X)1184 3706 y Fi(m;n)p
693 Ff(=0)1506 3541 y Fk(1)p 1426 3585 209 4 v 1426 3677
694 a(2)1475 3648 y Fi(n)1522 3677 y Fj(m)p Fk(!)1666 3608
695 y Fh(\001)1716 3462 y Fg( )1820 3541 y Fj(d)p 1792 3585
696 107 4 v 1792 3677 a(dx)1908 3462 y Fg(!)1974 3485 y Fi(m)2057
697 3608 y Fk(\()p Fj(x)2150 3567 y Ff(2)2212 3608 y Fh(\000)22
698 b Fk(1\))2398 3567 y Fi(n)2467 3608 y Fh(\001)g Fj(y)2569
699 3567 y Fi(m)2657 3608 y Fh(\001)g Fj(z)2756 3567 y Fi(n)257
700 3903 y Fk(Because)34 b(the)f(T)-8 b(a)m(ylor)31 b(series)i(dev)m
701 (elopmen)m(t)g(theorem)f(holds)f(in)h(formal)d(p)s(o)m(w)m(er)34
702 b(series)257 4024 y(rings)e(\(see)i([1],)f(section)f(2.16\),)g(w)m(e)i
703 (can)f(simplify)752 4307 y Fj(G)83 b Fk(=)1102 4199 y
704 Fe(1)1078 4224 y Fg(X)1071 4405 y Fi(n)p Ff(=0)1254 4240
705 y Fk(1)p 1231 4284 96 4 v 1231 4375 a(2)1280 4347 y Fi(n)1358
706 4307 y Fh(\001)1408 4161 y Fg( )1515 4199 y Fe(1)1490
707 4224 y Fg(X)1474 4405 y Fi(m)p Ff(=0)1685 4240 y Fk(1)p
708 1653 4284 113 4 v 1653 4375 a Fj(m)p Fk(!)1798 4307 y
709 Fh(\001)1847 4161 y Fg( )1951 4240 y Fj(d)p 1923 4284
710 107 4 v 1923 4375 a(dx)2039 4161 y Fg(!)2105 4184 y Fi(m)2188
711 4307 y Fk(\()p Fj(x)2281 4266 y Ff(2)2343 4307 y Fh(\000)23
712 b Fk(1\))2530 4266 y Fi(n)2599 4307 y Fh(\001)e Fj(y)2700
713 4266 y Fi(m)2766 4161 y Fg(!)2854 4307 y Fh(\001)h Fj(z)2953
714 4266 y Fi(n)912 4592 y Fk(=)1102 4484 y Fe(1)1078 4509
715 y Fg(X)1071 4690 y Fi(n)p Ff(=0)1254 4524 y Fk(1)p 1231
716 4568 96 4 v 1231 4660 a(2)1280 4631 y Fi(n)1358 4592
717 y Fh(\001)1408 4495 y Fg(\020)1458 4592 y Fk(\()p Fj(x)g
718 Fk(+)g Fj(y)t Fk(\))1761 4550 y Ff(2)1822 4592 y Fh(\000)h
719 Fk(1)1971 4495 y Fg(\021)2020 4518 y Fi(n)2089 4592 y
720 Fh(\001)f Fj(z)2188 4550 y Fi(n)912 4863 y Fk(=)1505
721 4795 y(1)p 1081 4840 898 4 v 1081 4932 a(1)g Fh(\000)1261
722 4892 y Ff(1)p 1261 4908 36 4 v 1261 4966 a(2)1323 4932
723 y Fk(\(\()p Fj(x)h Fk(+)f Fj(y)t Fk(\))1665 4903 y Ff(2)1725
724 4932 y Fh(\000)h Fk(1\))16 b Fj(z)257 5147 y Fk(W)-8
725 b(e)30 b(tak)m(e)f(o)m(v)m(er)h(the)g(terminology)c(from)i(the)h
726 (\\diag)p 2160 5147 30 4 v 34 w(rational")d(pap)s(er;)k(here)g
727 Fj(R)f Fk(=)e Fj(Q)p Fk([)p Fj(x)p Fk(])257 5268 y(and)h
728 Fj(M)38 b Fk(=)28 b Fj(Q)p Fk([[)p Fj(x)p Fk(]])g(\(or,)h(if)d(y)m(ou)i
729 (lik)m(e)f(it)f(b)s(etter,)j Fj(M)39 b Fk(=)27 b Fj(H)8
730 b Fk(\()p Fj(C)f Fk(\),)28 b(the)g(algebra)e(of)h(functions)257
731 5388 y(holomorphic)37 b(in)h(the)h(en)m(tire)f(complex)g(plane\).)62
732 b Fj(G)38 b Fh(2)g Fj(M)10 b Fk([[)p Fj(y)t(;)17 b(z)t
733 Fk(]])39 b(is)f(rational;)i(hence)1852 5637 y(1)p eop
735 2 1 bop 257 573 a Fj(F)45 b Fk(is)30 b(algebraic)g(o)m(v)m(er)i
736 Fj(R)q Fk([)p Fj(z)t Fk(].)44 b(Let's)31 b(pro)s(ceed)h(exactly)f(as)g
737 (in)g(the)g(\\diag)p 2891 573 30 4 v 34 w(series")g(pap)s(er.)257
738 693 y Fj(F)14 b Fk(\()p Fj(z)421 657 y Ff(2)461 693 y
739 Fk(\))33 b(is)f(the)h(co)s(e\016cien)m(t)g(of)f Fj(t)1399
740 657 y Ff(0)1471 693 y Fk(in)356 958 y Fj(G)p Fk(\()p
741 Fj(z)t(t;)610 890 y(z)p 610 934 50 4 v 617 1026 a(t)669
742 958 y Fk(\))c(=)1254 890 y(2)p Fj(t)p 848 934 895 4 v
743 848 1026 a Fk(2)p Fj(t)23 b Fh(\000)f Fk(\(\()p Fj(x)h
744 Fk(+)f Fj(z)t(t)p Fk(\))1428 997 y Ff(2)1490 1026 y Fh(\000)h
745 Fk(1\))16 b Fj(z)1780 958 y Fk(=)2598 890 y(2)p Fj(t)p
746 1894 934 1493 4 v 1894 1026 a Fh(\000)p Fj(z)2020 997
747 y Ff(3)2083 1026 y Fh(\001)22 b Fj(t)2168 997 y Ff(2)2229
748 1026 y Fk(+)g(2\(1)g Fh(\000)h Fj(xz)2689 997 y Ff(2)2729
749 1026 y Fk(\))f Fh(\001)g Fj(t)h Fk(+)f(\()p Fj(z)k Fh(\000)d
750 Fj(x)3259 997 y Ff(2)3299 1026 y Fj(z)t Fk(\))257 1234
751 y(The)34 b(splitting)c(\014eld)j(of)f(the)h(denominator)e(is)h
752 Fj(L)c Fk(=)f Fj(Q)p Fk(\()p Fj(x)p Fk(\)\()p Fj(z)t
753 Fk(\)\()p Fj(\013)q Fk(\))34 b(where)1168 1523 y Fj(\013)1230
754 1539 y Ff(1)p Fi(=)p Ff(2)1368 1523 y Fk(=)1481 1456
755 y(1)22 b Fh(\000)h Fj(xz)1756 1420 y Ff(2)1819 1456 y
756 Fh(\006)1918 1374 y(p)p 2001 1374 574 4 v 82 x Fk(1)f
757 Fh(\000)h Fk(2)p Fj(xz)2325 1427 y Ff(2)2387 1456 y Fk(+)f
758 Fj(z)2534 1427 y Ff(4)p 1481 1500 1094 4 v 1983 1592
759 a Fj(z)2032 1563 y Ff(3)834 1946 y Fj(\013)28 b Fk(=)f
760 Fj(\013)1089 1961 y Ff(1)1157 1946 y Fk(=)1290 1878 y(2)p
761 1270 1922 90 4 v 1270 2014 a Fj(z)1319 1985 y Ff(3)1391
762 1946 y Fh(\000)1501 1878 y Fk(2)p Fj(x)p 1501 1922 105
763 4 v 1528 2014 a(z)1637 1946 y Fk(+)1745 1878 y(1)22 b
764 Fh(\000)h Fj(x)1971 1842 y Ff(2)p 1745 1922 266 4 v 1854
765 2014 a Fk(2)2021 1946 y Fj(z)k Fk(+)22 b Fh(\001)17 b(\001)g(\001)25
766 b(2)k Fj(Q)p Fk(\()p Fj(x)p Fk(\)[[)p Fj(z)t Fk(]][)2831
767 1878 y(1)p 2831 1922 50 4 v 2831 2014 a Fj(z)2892 1946
768 y Fk(])1168 2368 y Fj(\013)1230 2383 y Ff(2)1297 2368
769 y Fk(=)1411 2300 y Fj(x)1466 2264 y Ff(2)1528 2300 y
770 Fh(\000)22 b Fk(1)p 1411 2345 266 4 v 1519 2436 a(2)1686
771 2368 y Fj(z)27 b Fk(+)22 b Fh(\001)17 b(\001)g(\001)26
772 b(2)i Fj(Q)p Fk(\()p Fj(x)p Fk(\)[[)p Fj(z)t Fk(]][)2496
773 2300 y(1)p 2496 2345 50 4 v 2496 2436 a Fj(z)2557 2368
774 y Fk(])257 2619 y(The)34 b(partial)c(fraction)i(decomp)s(osition)e(of)i
775 Fj(G)p Fk(\()p Fj(z)t(t;)2140 2580 y Fi(z)p 2140 2596
776 36 4 v 2145 2653 a(t)2186 2619 y Fk(\))g(reads)897 2893
777 y Fj(G)p Fk(\()p Fj(z)t(t;)1151 2825 y(z)p 1151 2869
778 50 4 v 1158 2961 a(t)1210 2893 y Fk(\))c(=)g Fh(\000)1487
779 2825 y Fk(2)p 1467 2869 90 4 v 1467 2961 a Fj(z)1516
780 2932 y Ff(3)1588 2893 y Fh(\001)1786 2825 y Fk(1)p 1648
781 2869 325 4 v 1648 2961 a Fj(\013)1710 2976 y Ff(1)1772
782 2961 y Fh(\000)22 b Fj(\013)1933 2976 y Ff(2)2005 2893
783 y Fh(\001)2055 2771 y Fg(\022)2204 2825 y Fj(\013)2266
784 2840 y Ff(1)p 2126 2869 259 4 v 2126 2961 a Fj(t)g Fh(\000)h
785 Fj(\013)2345 2976 y Ff(1)2416 2893 y Fh(\000)2604 2825
786 y Fj(\013)2666 2840 y Ff(2)p 2526 2869 V 2526 2961 a
787 Fj(t)f Fh(\000)h Fj(\013)2745 2976 y Ff(2)2794 2771 y
788 Fg(\023)257 3153 y Fk(It)33 b(follo)m(ws)e(that)656 3393
789 y Fj(F)14 b Fk(\()p Fj(z)820 3351 y Ff(2)860 3393 y Fk(\))28
790 b(=)f Fh(\000)1137 3325 y Fk(2)p 1116 3369 90 4 v 1116
791 3461 a Fj(z)1165 3432 y Ff(3)1238 3393 y Fh(\001)1436
792 3325 y Fk(1)p 1298 3369 325 4 v 1298 3461 a Fj(\013)1360
793 3476 y Ff(1)1421 3461 y Fh(\000)c Fj(\013)1583 3476 y
794 Ff(2)1654 3393 y Fh(\001)1704 3271 y Fg(\022)1861 3325
795 y Fj(\013)1923 3340 y Ff(1)p 1775 3369 273 4 v 1775 3461
796 a Fk(0)f Fh(\000)h Fj(\013)2008 3476 y Ff(1)2079 3393
797 y Fh(\000)g Fk(0)2228 3271 y Fg(\023)2317 3393 y Fk(=)2734
798 3325 y(1)p 2430 3369 657 4 v 2430 3387 a Fh(p)p 2513
799 3387 574 4 v 82 x Fk(1)f Fh(\000)h Fk(2)p Fj(xz)2837
800 3440 y Ff(2)2899 3469 y Fk(+)f Fj(z)3046 3440 y Ff(4)257
801 3660 y Fk(hence)1391 3800 y Fj(F)14 b Fk(\()p Fj(z)t
802 Fk(\))28 b(=)2019 3733 y(1)p 1735 3777 617 4 v 1735 3795
803 a Fh(p)p 1818 3795 534 4 v 81 x Fk(1)22 b Fh(\000)g Fk(2)p
804 Fj(xz)27 b Fk(+)22 b Fj(z)2311 3847 y Ff(2)404 4038 y
805 Fk(It)34 b(follo)m(ws)f(that)h(\(1)23 b Fh(\000)g Fk(2)p
806 Fj(xz)28 b Fk(+)23 b Fj(z)1581 4002 y Ff(2)1622 4038
807 y Fk(\))g Fh(\001)1761 3999 y Fi(d)p 1744 4015 72 4 v
808 1744 4072 a(dz)1825 4038 y Fj(F)37 b Fk(+)23 b(\()p Fj(z)28
809 b Fh(\000)c Fj(x)p Fk(\))g Fh(\001)e Fj(F)44 b Fk(=)31
810 b(0.)47 b(This)35 b(is)e(equiv)-5 b(alen)m(t)257 4158
811 y(to)33 b(the)g(claimed)d(recurrence.)404 4279 y(Starting)f(from)h(the)
812 h(closed)g(form)e(for)i Fj(F)14 b Fk(,)30 b(w)m(e)i(compute)f(a)g
813 (linear)e(relation)g(for)h(the)257 4399 y(partial)h(deriv)-5
814 b(ativ)m(es)32 b(of)g Fj(F)14 b Fk(.)44 b(W)-8 b(rite)31
815 b Fj(@)1637 4414 y Fi(x)1710 4399 y Fk(=)1843 4360 y
816 Fi(d)p 1823 4376 77 4 v 1823 4433 a(dx)1942 4399 y Fk(and)h(\001)2212
817 4414 y Fi(z)2280 4399 y Fk(=)c Fj(z)2461 4360 y Fi(d)p
818 2443 4376 72 4 v 2443 4433 a(dz)2525 4399 y Fk(.)44 b(One)33
819 b(computes)1674 4619 y Fj(F)41 b Fk(=)28 b(1)21 b Fh(\001)h
820 Fj(F)1273 4842 y Fg(\020)1323 4939 y Fk(1)f Fh(\000)i
821 Fk(2)p Fj(xz)k Fk(+)22 b Fj(z)1816 4898 y Ff(2)1856 4842
822 y Fg(\021)1928 4939 y Fh(\001)g Fj(@)2029 4954 y Fi(x)2073
823 4939 y Fj(F)42 b Fk(=)27 b Fj(z)g Fh(\001)22 b Fj(F)1209
824 5212 y Fg(\020)1258 5308 y Fk(1)g Fh(\000)h Fk(2)p Fj(xz)k
825 Fk(+)22 b Fj(z)1752 5267 y Ff(2)1792 5212 y Fg(\021)1842
826 5235 y Ff(2)1903 5308 y Fh(\001)g Fj(@)2009 5267 y Ff(2)2004
827 5333 y Fi(x)2049 5308 y Fj(F)42 b Fk(=)27 b(3)p Fj(z)2355
828 5267 y Ff(2)2417 5308 y Fh(\001)22 b Fj(F)1852 5637 y
831 3 2 bop 1089 697 a Fg(\020)1139 793 y Fk(1)21 b Fh(\000)i
832 Fk(2)p Fj(xz)k Fk(+)22 b Fj(z)1632 752 y Ff(2)1672 697
833 y Fg(\021)1744 793 y Fh(\001)g Fk(\001)1875 808 y Fi(z)1915
834 793 y Fj(F)41 b Fk(=)28 b(\()p Fj(xz)f Fh(\000)c Fj(z)2437
835 752 y Ff(2)2477 793 y Fk(\))f Fh(\001)g Fj(F)893 1066
836 y Fg(\020)942 1162 y Fk(1)g Fh(\000)h Fk(2)p Fj(xz)k
837 Fk(+)22 b Fj(z)1436 1121 y Ff(2)1476 1066 y Fg(\021)1526
838 1089 y Ff(2)1587 1162 y Fh(\001)g Fj(@)1688 1177 y Fi(x)1732
839 1162 y Fk(\001)1813 1177 y Fi(z)1853 1162 y Fj(F)42 b
840 Fk(=)27 b(\()p Fj(z)g Fk(+)22 b Fj(xz)2373 1121 y Ff(2)2436
841 1162 y Fh(\000)h Fk(2)p Fj(z)2634 1121 y Ff(3)2673 1162
842 y Fk(\))g Fh(\001)e Fj(F)650 1435 y Fg(\020)700 1532
843 y Fk(1)g Fh(\000)i Fk(2)p Fj(xz)k Fk(+)22 b Fj(z)1193
844 1491 y Ff(2)1233 1435 y Fg(\021)1283 1458 y Ff(2)1345
845 1532 y Fh(\001)f Fk(\001)1475 1491 y Ff(2)1475 1556 y
846 Fi(z)1515 1532 y Fj(F)42 b Fk(=)1723 1435 y Fg(\020)1773
847 1532 y Fj(xz)27 b Fk(+)22 b(\()p Fj(x)2091 1491 y Ff(2)2153
848 1532 y Fh(\000)h Fk(2\))p Fj(z)2389 1491 y Ff(2)2451
849 1532 y Fh(\000)f Fj(xz)2654 1491 y Ff(3)2717 1532 y Fk(+)g
850 Fj(z)2864 1491 y Ff(4)2904 1435 y Fg(\021)2976 1532 y
851 Fh(\001)g Fj(F)257 1764 y Fk(Solv)m(e)33 b(a)f(homogeneous)h(5)22
852 b Fh(\002)h Fk(6)32 b(system)h(of)g(linear)d(equations)j(o)m(v)m(er)h
853 Fj(Q)p Fk(\()p Fj(x)p Fk(\))f(to)f(get)443 1917 y Fg(\020)492
854 2013 y Fk(1)22 b Fh(\000)h Fk(2)p Fj(xz)k Fk(+)22 b Fj(z)986
855 1972 y Ff(2)1026 1917 y Fg(\021)1076 1940 y Ff(2)1137
856 2013 y Fh(\001)1187 1917 y Fg(\020)1237 2013 y Fk(\()p
857 Fh(\000)p Fk(2)p Fj(x)p Fk(\))g Fh(\001)g Fj(@)1617 2028
858 y Fi(x)1661 2013 y Fj(F)36 b Fk(+)22 b(\(1)g Fh(\000)h
859 Fj(x)2122 1972 y Ff(2)2162 2013 y Fk(\))f Fh(\001)g Fj(@)2328
860 1972 y Ff(2)2323 2038 y Fi(x)2368 2013 y Fj(F)36 b Fk(+)22
861 b(\001)2646 2028 y Fi(z)2686 2013 y Fj(F)36 b Fk(+)22
862 b(\001)2964 1972 y Ff(2)2964 2038 y Fi(z)3004 2013 y
863 Fj(F)3081 1917 y Fg(\021)3158 2013 y Fk(=)27 b(0)257
864 2233 y(Divide)32 b(b)m(y)h(the)g(\014rst)g(factor)f(to)g(get)864
865 2453 y(\()p Fh(\000)p Fk(2)p Fj(x)p Fk(\))23 b Fh(\001)f
866 Fj(@)1245 2468 y Fi(x)1289 2453 y Fj(F)36 b Fk(+)22 b(\(1)g
867 Fh(\000)h Fj(x)1750 2412 y Ff(2)1789 2453 y Fk(\))g Fh(\001)e
868 Fj(@)1955 2412 y Ff(2)1950 2478 y Fi(x)1996 2453 y Fj(F)36
869 b Fk(+)22 b(\001)2274 2468 y Fi(z)2314 2453 y Fj(F)35
870 b Fk(+)22 b(\001)2591 2412 y Ff(2)2591 2478 y Fi(z)2632
871 2453 y Fj(F)41 b Fk(=)27 b(0)257 2673 y(This)h(is)f(equiv)-5
872 b(alen)m(t)27 b(to)h(the)g(claimed)d(equation)j(\(1)12
873 b Fh(\000)g Fj(x)2293 2637 y Ff(2)2332 2673 y Fk(\))g
874 Fh(\001)g Fj(P)2499 2614 y Fc(00)2485 2698 y Fi(n)2543
875 2673 y Fk(\()p Fj(x)p Fk(\))g Fh(\000)g Fk(2)p Fj(x)g
876 Fh(\001)g Fj(P)3008 2614 y Fc(0)2994 2698 y Fi(n)3041
877 2673 y Fk(\()p Fj(x)p Fk(\))g(+)g(\()p Fj(n)3368 2637
878 y Ff(2)3419 2673 y Fk(+)257 2794 y Fj(n)p Fk(\))23 b
879 Fh(\001)e Fj(P)488 2809 y Fi(n)535 2794 y Fk(\()p Fj(x)p
880 Fk(\))28 b(=)g(0.)257 3126 y Fb(References)62 3345 y
881 Fk(1)146 b(Bruno)46 b(Haible:)69 b(D-\014nite)45 b(p)s(o)m(w)m(er)i
882 (series)g(in)e(sev)m(eral)i(v)-5 b(ariables.)82 b Fa(Diploma)46
883 b(thesis,)501 3466 y(University)35 b(of)g(Karlsruhe,)f(June)h(1989)p
884 Fk(.)43 b(Sections)33 b(2.14,)f(2.15)g(and)h(2.22.)1852
888 userdict /end-hook known{end-hook}if
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