ADVANCED PROBLEMS AND SOLUTIONS

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1 ADVANCED PROBLEMS AND SOLUTIONS EDITED BY FLORIAN LUCA Please sed all couicatios cocerig ADVANCED PROBLEMS AND SOLUTIONS to FLORIAN LUCA, SCHOOL OF MATHEMATICS, UNIVERSITY OF THE WITWA- TERSRAND, PRIVATE BAG X3, WITS 200, JOHANNESBURG, SOUTH AFRICA or by e-ail at the address as files of the type tex, dvi, ps, doc, htl, pdf, etc This departet especially welcoes probles believed to be ew or extedig old results Proposers should subit solutios or other iforatio that will assist the editor To facilitate their cosideratio, all solutios set by regular ail should be subitted o separate siged sheets withi two oths after publicatio of the probles PROBLEMS PROPOSED IN THIS ISSUE H-80 Proposed by D M Bătieţu-Giurgiu, Bucharest ad Neculai Staciu, Buzău, Roaia Prove that if 2, p 1 are itegers ad 0, x > 0 are real ubers for 1,,, the lettig X 1 x, we have the iequality (F p X +F p+1 x +1 (Fp+1 2 X Fpx (F p +F p (Fp+1 2 F2 p 2+1 X 1 H-806 Proposed by Hideyui Ohtsua, Saitaa, Japa The two sequeces {T } Z ad {S } Z satisfy T +3 T +2 +T +1 +T with T 0 0, T 1 T 2 1, S +3 S +2 +S +1 +S with S 0 3, S 1 1, S 2 3 for all itegers For 0 prove that T ( 2 S ( 2 T 2( 2 H-807 Proposed by Mehtaab Sawhey, Coac, NY ad Prove for positive itegers that i µ(gcd(i, j i i1 i1 j1 j1 µ(gcd(i, j ij φ(, 1 2 ω( VOLUME, NUMBER 2

2 H-808 Proposed by Mehtaab Sawhey, Coac, NY ADVANCED PROBLEMS AND SOLUTIONS Prove that /2 j0 ( j,j, 2j ( ( 2 1 3i ( 1 i i 1 /2 i0 SOLUTIONS A Itegral with the Gaa Fuctio ad Fiboacci Nubers H-771 Proposed by D M Bătieţu-Giurgiu, Bucharest ad Neculai Staciu, Buzău, Roaia (Vol 3, No 2, May 201 Let > 0 ad Γ : (0, (0, be the gaa fuctio Calculate +1 (+1! x li Γ(! F dx Solutio by Ágel Plaza We will show that f is a cotiuous real fuctio i (a,b ad α /e (a,b the +1 (+1! ( x li f! F dx 1 e f ( α I( our case, Γ is a cotiuous real fuctio i (0, ad therefore the required liit is 1 α e Γ e Let b +1 (+1! itegrals, +1 (+1!! f ( x F ad a! F dx F b f(t dt F a e The, by the Mea Value Theore for for soe t (a,b Now, by the Stirlig approxiatio forula, l(! l( + 1 ( 1 2 l(+l( 2π+O, so l (! l! F (b a f(t ( l l 1+O 1+o(1 as Thus, usig also the Biet forula for F which iplies that li F α, we have li b li a li t α e MAY

3 THE FIBONACCI QUARTERLY By the cotiuity of f at α /e, we have li F (b a f(t f ( α li e ( +1 (+1!! ( α! f li e 1 ( α e f e Also solved by Ditry Fleischa, Nicuşor Zlota, ad the proposers A Geoetric Iequality H-772 Proposed by D M Bătieţu-Giurgiu, Bucharest ad Neculai Staciu, Buzău, Roaia (Vol 3, No 2, May 201 If ABC is a oiscosceles triagle the prove that a 8 (bf 2 +cf+1 2 (a b2 (a c 2 > 288r3 3 F 2+1 perutatios Here, a,b,c,r are the legths of the sides ad the radius of the iscribed circle of the triagle ABC, respectively Solutio by the proposers By the Harald Bergströ iequality ad F 2 +F 2 +1 F 2+1, we have: W perutatios perutatios a 8 (bf 2 +cf2 +1 (a b2 (a c 2 ( 2 (a b(a c bf 2 +cf (a+b+c(f 2 +F (a+b+cf 2+1 The su i paretheses siplifies to perutatios perutatios ( perutatios perutatios perutatios 2 (a b(a c (bf 2 +cf 2 +1 (a b(a c (a b(a c (a b(a c a4 (b c b 4 (c a c 4 (a b (a b(b c(c a a 2 +b 2 +c 2 +ab+bc+ca 186 VOLUME, NUMBER 2 2 2

4 ADVANCED PROBLEMS AND SOLUTIONS Sice a 2 +b 2 +c 2 ab+bc+ca 4S 3, we get 1 W (8S S2 192(pr2 96pr2 288r3 3, (a+b+cf 2+1 2pF 2+1 2pF 2+1 F 2+1 F 2+1 where for the last iequality we used the fact that p 3 3r Rear The iequality is strict because ABC is ot equilateral A Su with Bioial Coefficiets, Fiboacci ad Beroulli Nubers H-773 Proposed by H Ohtsua, Saitaa, Japa (Vol 3, No 3, August 201 Let B be the Beroulli ubers defied by the geeratig fuctio x e x 1 For itegers 0 ad 0, prove that ( [ 2 F 2 B 2 2( 2 Solutio by the proposer It is ow that L r1 B! x (α r 2 1 +L (2 1 ] B (x+1 B (x x 1, where B (x ( B x By this idetity, we have 2 ( 2 ((α r +1 (α r B 2 2(α r 2 1 Usig this idetity, we have L r1 2 2(α r 2 1 ( 2 ( ( { } L 2 ((α r+1 (α r B 2 r1 2 ( 2 (α (α L B 2 (α 2 β 2 B 2( + ( 2 2 ( F 2 B 2( L (2 1 Therefore, we obtai the desired idetity Also solved by Ditry Fleischa (α ( β B 2 (α (2 1 +β (2 1 B 1 MAY

5 THE FIBONACCI QUARTERLY Bessel Fuctios with Fiboacci ad Lucas Nubers H-774 Proposed by G C Greubel, Newport News, VA (Vol 3, No 3, August Let 0, p 0 be itegers Evaluate the series F +p L + (+p!(+! i ters of the Bessel fuctios 2 Evaluate the case p i ters of a series of odified Bessel fuctios of the first id Tae the liitig case 0 3 Show that whe p 0 the series is give by F L +!(+! 1 (I (2α I (2β F J (2 Solutio by the proposer Part 1 Let the series i questio be give by Sp F +p L + (+p!(+! Without uch difficulty it is see that F +p L +p F 2+p+ +( 1 + F p Use of this expressio leads the series S p to the for S p F 2+p+ (+p!(+! +( 1 F p This curret expressio ca be ore easily see i the for where ( 1 (+p!(+! S p 1!p! ( α p+ f(α 2 ;p, β p+ f(β 2 ;p, + ( 1 F p f( 1;p,, (1!p! f(x;p, x (p+1 (+1 (2 The series give by f(x;p, is of the hypergeoetric type 1 F 2 ad ca the be related to the Loel fuctios, which are of the Bessel faily of fuctios The Loel fuctios are expressed by s µ,ν (z z µ+1 ( F (µ ν +1(µ+ν ; µ ν +3, 2 µ+ν +3 ; z VOLUME, NUMBER 2

6 ADVANCED PROBLEMS AND SOLUTIONS Whe µ ad ν are set to the values µ p+ 1 ad ν p the Loel fuctio reduces to s +p 1, p (z z+p F 4p 1 2 (1;p+1,+1; z2 4 Upo aig the chage of variable z 2i x it is see that s +p 1, p (2i x 2+p 2 i +p x (+p/2 Copariso of equatios (2 ad (3 lead to p f(x;p, (p 22 p ( i +p With this result equatio (1 becoes x (+p/2 1F 2 (1;p+1,+1;x (3 s +p 1, p (2i x S p ( i+p 2 2 p Γ(Γ(p [s +p 1, p (2iα s +p 1, p (2iβ] + ( 1p 2 2 p F p Γ(Γ(p s +p 1, p ( 2 (4 As a alterate for the odified Loel fuctios ca be used, give by (see paper [1] ad the refereces therei: z µ+1 ( µ ν +3 µ+ν +3 t µ,ν (z F (µ ν +1(µ+ν ;, ; z2, ad have the relatio t µ,ν (x ( i µ+1 s µ,ν (ix With this, equatio (4 becoes S p 2 2 p Γ(Γ(p [t +p 1, p (2α t +p 1, p (2β] + ( 1p 2 2 p F p Γ(Γ(p s +p 1, p ( 2 The desired relatio sought is, or equatio (4, F +p L + (+p!(+! 2 2 p [t +p 1, p (2α t +p 1, p (2β] Γ(Γ(p + ( 1p 2 2 p F p Γ(Γ(p s +p 1, p ( 2 Part 2 Loel s fuctio ca be expaded i ters of a series ivolvig the Bessel fuctio of the first id Whe µ±ν 1, 2, it is give that (see equatio 1197 i [2]: s µ,ν (z 2 µ+1 (2 +µ+1γ( +µ+1!(2 +µ ν +1(2 +µ+ν +1 J 2+µ+1(z Whe z ix, the Bessel fuctio becoes the odified Bessel fuctio of the first id ad is give by J (ix i I (x, the result is s µ,ν (ix (2i µ+1 ( 1 (2 +µ+1γ( +µ+1!(2 +µ ν +1(2 +µ+ν +1 I 2+µ+1(z MAY

7 THE FIBONACCI QUARTERLY Whe µ p+ 1 ad ν p this becoes (2i +p 2 ( 1 (2 ++pγ( ++p s +p 1, p (ix!( +p( + I 2++p (x Maig use of this relatio equatio (4 becoes S p 1 Γ(Γ(p ( 1 (2 ++pγ( ++p!( +p( + [ I 2++p (2α I 2++p (2β+ ( 1 +p F p J 2++p ( 2 Whe p this reduces to F 2+2 [(+!] 2 2 Γ 2 ( ( 1 Γ( +2!( + [I 2+2 (2α I 2+2 (2β] or F 2+2 [(+!] 2 ( 2 ( 1 (2!( + [I 2+2(2α I 2+2 (2β] This is the desired result of Part 2 It ay be oted tha whe 0 the expressio ca be reduced to F 2 [(!] 2 1 [I 0 (2α I 0 (2β] ( ] Part 3 Sice F L + F 2+ ( 1 F it ca be easily see that F L +!(+! F 2+!(+! F [ 1 α F α 2!(+! β ( 1!(+! ( 1!(+! F L +!(+! 1 (I (2α I (2β F J (2, β 2!(+! where J (x ad I (x are the Bessel ad odified Bessel fuctios of the first id, respectively Whe 0 this result reproduces ( Fro the relatio F +p L F 2+p +( 1 p F p it follows that F +p L!(+p! 1 (I p (2α I p (2β+F p J p (2 ] 190 VOLUME, NUMBER 2

8 ADVANCED PROBLEMS AND SOLUTIONS Refereces [1] C H Zeier ad H P Schleer, The iverse Laplace trasfors of the odified Loel fuctios, Itegral Trasfors ad Special Fuctios, 242 (2013, [2] Digital Library of Matheatical Fuctios, DLMF, Also solved by Ditry Fleischa MAY

ADVANCED PROBLEMS AND SOLUTIONS

ADVANCED PROBLEMS AND SOLUTIONS EDITED BY FLORIAN LUCA Please sed all commuicatios cocerig ADVANCED PROBLEMS AND SOLUTIONS to FLORIAN LUCA, SCHOOL OF MATHEMATICS, UNIVERSITY OF THE WITWA- TERSRAND, WITS