<aj + ck) m (bj + dk) n. E E (-D m+n - j - k h)(i) j=0 k=0 \ / \ /

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1 ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND j=. WHITNEY Loc Have State College Loc Have # Pesylvaia Sed all commuicatios cocerig Advaced Problems ad Solutios to Raymod E. Whitey, Mathematics Departmet, Loc Have State College, Loc Have, Pesylvaia This departmet especially welcomes problems believed to be ew or extedig old results. Proposers should submit solutios or other iformatio that will assist the editor. To facilitate their cosideratio, solutios should be submitted o separate siged sheets withi two moths after publicatio of the problems. H-227 Proporsedby L Carlitz, Due Uiversity, Durham, North Carolia. Show that m E E (-D m+ - j - h)(i) j=0 =0 \ / \ / <aj + c) m (bj + d)! = mii mi(m,) (^(jja^^ek.) r=0 I particular, show that the Legedre polyomial P (x) satisfies (t) 2 P (x) = J2 ( - )3 (aj + C)R(bi + d)r "*(^)() j,=0 where ad = ^-(x + ), be = - ( x - ). H-228 Proposed by R. Whitey, Loc Have State College, Loc Have, Pesylvaia. F,, Defie the sequece M L ( uj ) as follows; =i u ~ (F ) ( > ), where F deotes the Fiboacci umber. 50

2 502 ADVANCED PROBLEMS AND SOLUTIONS [Dec. (). Fid a recurrece relatio for \ u j. ad (2). Fid a geeratig fuctio for the sequece, \ u / _-. H-229 Proposed by L Carliiz, Due Uiversity, Durham, North Carolia. A triagular array A(,) ( 0 < < ) is defied by meas of ( A( +, 2) = A(, 2 - ) + aa(, 2) (*) < I A( +, 2 + ) = A(, 2) + ba(, 2 + ) together with A(0,0) ~, A(0,) = 0 ( + 0). Fid A(,) ad show that ] A < > 2)(ab) = a(a + b ) ~ \ ^ A(, 2 + l)(ab) = (a + b) - SOLUTIONS ARRAY OF HOPE H-95 Proposed by Verer Hoggatt, Jr, Sa Jose State Uiversity, Sa Jose, Califoria Cosider the array idicated below: (i) Show that the row sums are F 0, > 2. 2 (ii) Show that the risig diagoal sums are the covolutio of ^ - l C o a d (uco; 2, 2 ) } ^,

3 973] ADVANCED PROBLEMS AND SOLUTIONS 503 the geeralized umbers of Harris ad Styles. Solutio by L Carlitz, Due Uiversity, Durham, North Carolia. Let A(,) deote the elemet i the row ad colum. The (presumably) A ( ' X ) = F 2-3 ( > } ad [A(, 2) = A(, 2 - ) + A( -, 2) [A(, 2 + ) = A( -, 2 + ) + A( - 2, 2) ( > ) Put F(x,y) = X ^ A (, =l 2 2 > xt V G < x >y> = 2 ] C =l A ( > 2 + )xr y 2+l OO The A(x) = ^ A (, l ) x. =l A(x) = x + F 2. 3 x = x + x ^ F ^ ^ x =2 =l = X + X x - x 2 x - 2x 2-3x + x^ - 3x + x^ Next F(x,y) = ]jt ^ 2 (A(, 2 - ) + A( -, 2 ) ) x y = xf(x,y) + yg(x,y), () ( - x)f(x,y) = yg(x,y). Also 4

4 504 ADVANCED PROBLEMS AND SOLUTIONS [Dec. G(x,y) = y ^ A (, l ) x 2+l + ^ ] (A( -, 2 + ) + A( - 2, 2)) x =2>0 = ya(x) + x J2J2 A( ' 2 + )xily + x2 yj2j2 = l > 0 = l > 0 A( ' 2^y 2 = y(l - x)a(x) + xg(x,y) + x 2 yf(x,y), (2) ( - x)g(x 5 y) = x 2 yf(x,y) + x ( " x ) ( " 2 x ) y. - 3x + x 2 It follows from () ad (2) that (3) Hece (( - x)» - x V ) F ( x, y ) = x ( - x ) ( - - 3x + x 2 2 x ) y 2 (( - x) 2 - v2 2\f x 2 y 2 )G(x,y) v,a = x(l - x) 2 (l - 2x)y - 3x + x ; (4) (( - x)* - x V ) T T A(,)x y = x? ( ~ x ) ( ~ 2 x ) ( " x + ^ ^"r i - 3x + x 2 For y = this reduces to x A<,) x(l - - x)(2 - x) 3x + x 2 X + - X - 3x + x z x + 2>2 x ' 2 A(,) = F 2 ( > ). If we tae y = x, Eq. (4) reduces to

5 973] ADVANCED PROBLEMS AND SOLUTIONS 505 OO Vx y>(- 5 ) = =2 x2( - ^ - 2 x > ( - x - x 2 )(l - x + x 2 )(l - 3x + x 2 ) S F 2-l x t l x(l - 2x) ( - X - X 2 ) ( l - X + X 2 ) This expresses the risig diagoal sums - 2 A( -, ) =l as covolutios as stated. Remar. It follows from (3) that 2-2 w \ - 2x \~^ x y F(x ' y) =, + 2 ^ ^ fci - 3x + x?. ( - x) ^ 2-l 2- ~/ v - 2x \ ^ x y G ( x, y ) = ^ _ ^ - ^ 2-3x + x^. ( - x) (5) E A(.2)*> = ( " 2 x ) x 2 " ' 2 i T \ ( - 3x + x 2 )(l - x ) 2 ^ =2-l ( " 2 X ) x A(, 2 - l)x = t T i ( - 3x + x 2 )(l - x) 2-2 =2-l By meas of (5) we ca obtai explicit formulas for A(,). Sice - 2x _ V 2 ^ F2r- T, x : r - 3x + x 2 r = Q OO it follows that Therefore, oo oo oo =2-l r=0 s=0 s X

6 506 ADVANC3ED PROBLEMS AND SOLUTIONS [Dec. -2+l r=0 V ' Similarly -2+l A(,2-) = (V-'s'Kr-l r=0 ' ( > " / t o solved by the Proposer. PARTITION H-96 Proposed by J. B. Roberts, Reed Collage, Portlad, Orego. (a) Let A 0 be the set of itegral parts of the positive itegral multiples of T, where T = + ^ 2 ' ad let A +» m = 0,, 2,, be the set of itegral parts of the umbers T 2 for c~ A. Prove that the collectio of Z of all positive itegers is the disjoit uio of the A.. J (b) Geeralize the propositio i (a). Solutio by L Carlltz, Due Uiversity, Durham, North Carolia.. Put a() = [T], b() = [T 2 ] = [(T + )] = a() +. Also for brevity put (a) = (a() =, 2, 3, }, (b) = {b() =, 2, 3, }. It is well ow that* (*) Z + = (a) U (b). Put (b a) = {b a() =, 2, 3, },

7 973] ADVANCED PROBLEMS AND SOLUTIONS 507 where juxtapositio deotes compositio. The it follows at oce from (*) that ad so o. Clearly this implies ad put Z + = (a) U (ba) U (b 2 ) = (a) U (ba) u (b 2 a) U (b 3 ), oo Z + = U (b a) = U A,. K =0 =0 2. Let a,fi be positive irratioal umbers such that The it is well ow that where, as above, Hece ad so o. oo (/a) + (l/j3) = a() = [a], b() = [j3]. Z + = (a) U (b), (a) = {a() =, 2, 3, }, b() = {b() =, 2, 3, }. Thus Z + = (a) U (ba) U (b 2 ) = (a) U (ba) U (b 2 a) U (b 3 ), Remar. Z + = U (b a). =0 The fuctios a(), b() i are studied i cosiderable detail i the paper by L. Carlitz, V. E. Hoggatt,, J r., ad Richard Scoville: "Fiboacci Represetatios/' Fiboacci Quarterly, Vol. 0, No., pp Also solved by the Proposer. Editorial Note: See Beatty's Theorem (America Math. Mothly, 33 (926), 59, ad34 (927) 59.) The editor wishes to acowledge solutios to H-94 by L. Frohma, P. Brucma, ad J. Ivie. Editorial Note: The followig list represets previous problem proposals (less tha or equal to H-00) which, to date, have ot bee solved: 22, 23, 40, 43, 46, 60, 6, 73, 76, 77, 84, 87, 90, 9, 94, ad 00. Startig i the ext sectio, we shall re-ru some of these proposals. ERRATA O Problem H-28, April, 973, please chage the matrix to read: J x