THE EVALUATION OF CERTAIN ARITHMETIC SUMS

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Morris Thornton
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1 THE EVALUATION OF CERTAIN ARITHMETIC SUMS B.C. GRIIVISON Dept. of Biostatistfes, Univesity of Woth Caolina, Chapel Hill, Woth Caolina In this pape we evaluate cetain cases of the expession (1.1) ^ max(ai,a 2.",A n ); whee ^k = aki + ~' + akm k whee the sum is ove all the a'%, each anging fom zeo to some positive intege. We also conside analogous sums fo min. Fo example we obtain, fom some geneal esults which we establish, the fomula, ] P max(a + h,c+d) = 22^ \ ) + 170( 2 ) + 42ol 3 ^ + 420^ 4 ] + 148^ ]. a,b,c,d=q Some geneal popeties of moe geneal cases of (1.1) ae established. Solutions of many poblems, paticulaly combinatoial poblems, ae often expessed in tems of such sums. Fo example, without going into detail, we fequently encounte sums of max min in poblems of enumeating aays. See H. An eta/. [1] Calstz [2] fo details. In a elated wok, Calitz [3] [4] obtains, elates to othe poblems, geneating functions fo max (ni, n^) min (n /,, n/<l Moe geneally, he evaluates M (ni. ~.nk) x -4 k (= I ~,ki, whee M (n^, n^) is defined by the following two popeties: (a) it is symmetic in n 1f -, % ; (b) if /?? < n 2 < < /? then He also evaluates the elated seies JOO MJ n V '"* n ki = n ( = t '",k) - Z x ^ - x ^ z ^ ' " ' " * ' (=1,..;k). nu'",nuo Roseile [6] examines the elationship between this seies the Euleian function. Othe than [3], [ 4 ], [6] this pape, thee appaently has been vey little published on poblems of this natue. A numbe of techniques ae employed to solve vaious aspects of the poblem compute computation was necessay in some instances. The main esults of this pape ae (3.3), (4.3), (4.7), (4.8), (4.11), (4.12), (4.13), (4.14), (4.15), (5.7), (5.9), (5.10), (5.131,(5.14), (5.15) (5.16). 2. Peliminay to ou discussion we need some basic popeties of Euleian polynomials. The n th Euleian polynomial, a n M, is defined,following Riodan [5],by Fom this definition we get ajx) = (1-x) nh J ] ^xk ' k=0 373

2 374 THE EVALUATION OF CERTAIN ARITHMETIC SOIVSS [DEC. a n (x) = nxa n / (x) + x( 1 - x)da n / (x), whee D is the diffeential opeato. Hence, the fist few polynomials, which we will use late, ae ao(x) = 1, aj(x) = x, a2(x) = x 2 +x, a 3 (x) = x 3 + 4x 2 +x, a 4 (x) = x 4 +11x 3 +11x 2 +x. A ecuence a table fo the coefficients (Euleian numbes) of Euleian polynomials a geneating function fo a n (x) may be found in [5; pp. 39, 215]. As fo convenient notation we wite max.. min (a ' b> > whee we wish to discuss both max (a,b) mm (a f b). Also we adopt the convention 3. Taking the simplest case fist we evaluate the sum To do this, we put 0 = <P = (1-X 1 )(l-x 2 )"(1-X n ). ]C nf)in(i v -J n ). (3.1) F(x u -.,x n ) = ] T which becomes l J2" min (i U "J n )xi -x n n, F(x 1f -,x n) = <f 1 J2 min ('1' '"' 'n> x 'l '" x 'n Now since mintiw-in) = 1= J2 2 j=0 k x +i+1=i x 1f it follows immediately that Theefoe, V* / - i h in A~1 x 1 x 2' x n > M minhi,-, n )xi -x = 0 jz s.. n. t-i (1-x 7 x 2 -x n ) tt «\ i-/ i M-2 x 1 x 2 ' x n (3.2) F(x h -,x n ) = 0 ^. t-xfx 2 '"X n Compaing the coefficients in the expansion of (3.2) with those of (3.1) gives (3.3) Ys min i<'"' M = ii* m ",in= 0 j<min( 1,, x%) i=1 If i =... = n = then the ight side of (3.3) educes to the familia seies n ( i -# ' 4. The sum 1=0

3 1874] THE EVALUATION OF CERTAIN ARITHMETIC SUMS 375 / "i'""' / "n (4.1) Y^ maxtii,-,i n ) is appaently moe difficult to evaluate. If n = 2 we may use (3.3) the identity (4.2) min (ij) + max (i,j) = i +j to get,$ min(,s) (4.3) ^ max (ij) = %( + 1)(s +D+ s(s + 1)( + 1)- J^ ( ~ J,(s " j ) ' i,j=0 l=q Now, consideing the case of (4.1) whee all the f% ae equal we define (4.4) F n () = ^ max ( '1< - / U <t,'~'in = 0 Then by an inclusion-exclusion agument, F n () = [ " ) 2 mdx ( '1' ""' ' n ~ 1 ' ) " ( 2 ) Z) max (i 1' '"' in ~ 2 ' ' ) o (- i+'fymaxfo-sf + Fnf-V (4.5) / V i W - ' X [ n k)(-1) k+1 (+1) n - k + F n (-D (>1). The expession (4.6) i K n k y-1) k + 1 (+1) k=1 k=1 may be simplified, accoding to the binomial theoem, giving so that (4.5) becomes (+W-t" (4.7) F n () = ( + 1) n - ^ + F n ( -1) (>1). Applying (4.7) to F n (-1) we get Fj) = f+i-/" 1 +(-.1) n -(-1) n+1 +F n (-2) continuing in this manne we eventually aive at -7-7 F n () = ] f-juf+7-#"_] f-w 1 *'. n-1 k=0 k=0 k=0 k=0 j=0 Now by compaing the last expession with (4.4) we have (4.8) F M = max 0 h -,i n ) = E ( /) ^ ix'jxo k=0 J =0

4 378 THE EVALUATION OF CERTAIN ARITHMETIC SUMS [DEC. o, in the usual notation fo Benoulli polynomials, (4.9) ± ^,,,..,,J«f^ /i,-,/ -0 1=0 The fist few special cases of F n (), obtained fom (4.7) ae F n (1) = 2 n -1, F n (2) = 2*3 n -2 n - 1, F n (3) = 3-4 n - 3 n " ~ 2 n ~ 1. Next we evaluate the geneating function (4.10) F n ()x n. n=0 Fom (4.8), (4.10) becomes n-1 n E Z{j\ E(7) E **v ^ ^ - E E (7) E ^'-"-E (") E '"^ n=0 j=0 k=0 n=0 j=0 v ' k=0 n=0 ' ' k=0 /- / E * E *" I) ( n ; y >^'- * to =0 n=0 y=0 A=0 /?=0 k=0 n=0 k=0 = 23 k(1-kx-x 7 -J2 k=0 k=0 k (1-kx)~ 1 We have theefoe poved ~ x J2 Ml - (k + 1)x)- 1 (1 - kx)" 1. k=0 (4.11) FjJx" = J2 I ] max(i v -J n )x n = x ] T k(1 - (k+1)x)~ 1 (1 - kx)~ 1. n=0 n=0 i lf -" f i n =0 k=0 Fom (4.11) it is easy to see that 2 ^ ^ ^ / _ K ; Z-, (i- { k+1)x)(1-kx) n,=0 k=0 If, on the othe h, we define Gn<y) = E =0 F n<)y then using the ecuence (4.7) setting F n (-1) = 0 we have G n M = E ((+ T) n ->" +1 )y + Y, F n< - Vv - E ^ =0 /=0 =0 V- n+1 )y +y E ^W^ = 0

5 1974] THE EVALUATION OF CERTAIN ARITHMETIC SUMS Then (1-y)G n (y) = (( + 1) n -^+1 )y = ^ (» V V ~ ' " + V =0 =0 k=0 =0 k=0 =0 =0 = y* ( n \ a k+1 (y) _ a n+1 (y) _ y ^ / V a k+1 (y) to {k > (1- Y ) k+2 (1- Y 2 to [kj n~v) k+2 ' whee 3pM. is the p Euleian polynomial intoduced in Section 2. Theefoe, we have (4.12) Gn<v) = Y* J2 ^^-wv/ -E( n-1 /i \ ^ - f / W - 0^khl-y) k+3 Fom (4.12) fom the list of Euleian polynomials in Section 2 we easily aive at the special cases G 2 (y)=3j Lt lf_ G 3 (y) = 7 Y +1 V 2 + Y 3, G 4 (y) = 1JY + 55y y 3 + y 4 _ (1-y) 4 (1-y) 5 (1-y) 6 The expansions of each of these geneating functions yield, fo > 0, (4.13) max(a,b) = 3[ \-{ \, a,b=0 (4.14) max(b,b,c) = 7{ Ywy Y\ V) (4.15) max(a,b,c,d)=15[ )+55[ ) + 25 { +. 2 ) + [ ), a,b,c,d=0 espectively a In geneal, fom (4.9) it may be seen that F n () is a polynomial in of degee n In this section, we conside A(,m,n) B(,m,n), whee (5.1) A(,m,n) = ^ max(ai + -+a m,b b n ), a u "^am^u"'b n =0 (5.2) B(,m,n) = ^ min (at+ -+a m,b b n ). a x,~',am,b l,-,h n =0 St is convenient to let the expession a,b=0 mean ai,,a m,bi,,b n = 0. Using the fomula (5.2) becomes max ',(ajb) = %(a+b±\a-b\), min

6 378 THE EVALUATION OF CERTAIN ARITHMETIC SUIVBS [DEC. (5.3) B(,m,n) = 1 A J2 hi+ - + a m + b 1 + >» + b n )-% J^ \ a i + '" + a m- b i-"'~ b n\ a f b=0 a,h=q SMow a,b=0 (5-4) i C a 1 + "' +a m"b 1 b n = ~J^( a 1 + ~+ a m-hi b n ) +Yl (a 1 + ~' + a m " b l - b n ), a,b=0 I II whee / // st fo ai + + a m K bj + -" + b n < max(m,n) bi b n < ai a m < max (m,n). espectively, whee it is undestd that a;, bf <. Moeove, maxim,n) J2 fa i + '" + a m-bf b n ) = J^ J2 ic (a a m -k) I k=0 b t +-+b n =k a x +-+a m <k b,-< a;< max (m,n) k - Z E E E u-k). k=0 j=0 b t +~'+b n =k a x +-+a m =i b/< a,-< Now let P(a,b,c) be the numbe of patitions of a > 0 into at most b pats, each pat <c, let P(O f b,c) = t maxim, n) k (5.5) ^hf + '-' + Hm-bf b n )= 2^ J^ (k-imkmmim) - / k=0 j=0 By a simila agument we find maxim,n) k (5.6) Yj (a i + '" + a m-bf- --b n ) = J^ 2 (*-im k > m >> p 0>s)- N k=0 j=0 By substituting (5.5) (5.6) into (5,4) efeing to (5.3), then by applying this same agument fo we find that A(,m,n) B(,m,n) ae given by (5-7) E ZZ(», + +»n,.b, + "-H>*)- (a+ak![ +VmH ' A(.mji) maxim,n) k k=0 j=0 espectively. Mow, although (5.7) expesses ou poblem in tems of a difficult patition poblem the fomula is nevetheless vey useful in that it affods us a method of detemining the geneating function. Hee we obseve, fom (5.7), that A(,m,n) B(,m,n) ae polynomials in. Futhemoe, thei degees ae less than o equal to m + n + 1, fo if not, then fo lage values of, B(,m,n) will be negative. In fact, in view of special cases, we may conjectue that the degee of A(,m,n) B(,m,n) is pecisely m+n + 1. We may now evaluate seveal cases. Fist, we conside B(,2,1).

7 1974] THE EVALUATIOW OF CERTAIN ARITHMETIC SUMS BtaV- min( a +b,c) = Y. < a+b > + E Afte some manipulation it is seen that C ~ E a,h,c=0 a+b< ~ x ^ G _ a+b>c -LU^ c=0 /! -_,_u- a+b=c a,b Fc< a,b,c< E c E ^ = ZZ; E *-J2> 3 ** 2 *A; a * 6 ^ c c=0 =0 a*/?=/ c=0 a A c < ^ Then we have E E *-,*+» c=0 a+b=c c=0 min(a+b,) B<,V) = Y[l- k 3 -U)+Y,c=T 2 2(+1 > (+1>+?, E ' c=0 a+b>c a,b=0 c=0 a,h,ck Since the degee of B(,2,1) is at most , it suffices to compute B(,2,1) fo = /, 2, J, 4. If we put so that min(a+b f ) «= E E «a,b=0 c=o (5.8) B(,2,1) = J- 2 ( + 1) (+ 1) + K Iz o then it is easy to compute Kj = 3, K 2 = 20, Ks = 71 K 4 =185. Then fom (5.8), we have Then we find the diffeences B( 1,2,1) = 3, B(2,2,1 )= 22, B(3,2,1) = 81 B(4,2,1) = 215. AB(,2,1) = 3, A 2 B(,2,1) = 16, A 3 B(,2,1) = 24, A 4 B(,2,1) =11. Substituting these values into IMewton's expansion fo the geneating function we have J] B(,2,1)x k = T A k B(0,2,1) ^- = ^ (3 + 7x+x 2 ). Upon exping we get (5.9) B(,2,1)= min(a + b,c) = 3[ )+7[ ) + { Fom (4.2), which, fom (5.9) educes to ^ P (max (a +b,c) + min (a +b f c)) = Y ^ (a+h+c) (5 1Q) V ^ m. a x<a / j + b,c). A -I = - o( + 3\_ 3{' 3 4 )+7[' 2 4 )+[' 4 iy 3 L7 l + 2 \,,f + 1\,3 / ^ i\3 :<+i

8 380 THE EVALUATION OF CERTAIN ARITHMETIC SUMS [DEC. Next we put B(,2,2) = Bf) fo bevity, conside (5.11) Bf) = 5Z min(a+b,c + d). Fom (5.7) we have a,h,c=q 2 k Bf) = ( + 1) 4 - ] T (k-j)pfk,2,)p(j,2,). k=0 j=0 Hence Bf) is a polynomial in of degee <5. Theefoe the geneating function is 5 E BfJS = T A l B(0) -0 1=0 (7-xJ x '. Resoting to a compute to calculate A'B(O) fo / = 1, 2, 3,4, 5, (5.12) becomes Exping fo the coefficients we find Y\ B()x - 10* + 90x x x x 5 ~ (f^x)2 ( f _ x )3 (1_ x) 4 (1_ x) 5 (1_ x) 6 ' (5.13) YJ a,b,c,d=0 min (a+h f c+d) In the vey same way that we obtained (5.13) we get = 70 ( 7 ] + 90\ 2 ) +24o[ 3 \ ( 4 ) +92 (5.14) ] T max(a+b,c+d) = 221 f \ \ +42ol 3 \ + 42o(^\ a,h fc,d= I A (5.15) J^ min (a+b +c,d) = 7 ( ; j + 61 ( ) ( ) ( 4 ) * 5 ( ] a,b,c,d=0 (5.16) ] T max(a+h+c,d) = IS^'A ( ^ ]+501 f a,b,c,d=0 3 )+508^ ^) ( ]. 6. The autho wishes to expess his appeciation to Pof. L. Calitz fo suggestions which esulted in simplifications of seveal aeas of this wok to Woden J. Updyke, J. fo the pogamming computing necessay in establishing (5.13), (5.14), (5.15), (5.16). The computations wee pefomed on an IBM 360 model 20 at the Electonic Compute Pogamming Institute, Geensb, Noth Caolina. Compute time was povided by the Institute. REFERENCES 1. Hash An, Vishwa Che Dumi Hansaj Gupta, "A Combinatoial Distibution Poblem/' Duke Math, J Vol. 33 (1966), pp L Calitz, "Enumeation of Symmetic Aays," Duke Math. J., Vol. 33 (1966), pp L Calitz, "The Geneating Function fo max fn 1f -, n k ),"Potugaliae Mathematics, Vol. 21 (1962), pp L Calitz, "Some Geneating Functions," Duke Math. J Vol. 30 (1963), pp J. Riodan, An intoduction to Combinatoial Analysis, New Yok, D.P. Roselle, "Repesentations fo Euleian Functions,"Potugaliae Mathematica, pp AAAAAAA

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506

H.W.GOULD West Virginia University, Morgan town, West Virginia 26506 A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone