G E N E R A L I Z E D C O N V O L U T I O N A R R A Y S

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1 G E N E R A L I Z E D C O N V O L U T I O N A R R A Y S V. E. HOGGATT,JR. San Jose State University, San Jose, California G. E. BERGUM South Dakota State University, Brookings, Sooth Dakota Let {*nfel 1. IWTRODUCTiON Kfo be any two sequences, then the Cauchy convolution of the two sequences is a sequence j c n I ^ given by the rule n (1.1) c n = Y^ kb n -k+l. whose terms are When we convolve a sequence with itself n times we obtain a new sequence called the /? t/7 convolution sequence. The rectangular array whose columns are the convolution sequences is called a convolution array where the n th column of the convolution array is the (n - 1) st convolution sequence the first column is the original sequence. In Figure 1, we illustrate the first four elements of the convolution array relative to the sequence iu n l^=1 " l U 2 "*' " 4 "\ 2u x u 2 2u x u z +u\ 2u x u 4 + 2u 2 u d u\ 3u\u 2 3u\u 2 + 3u x u\ 3u*u 4 +6u l u 2 u 2 +u 2 3 u; 4u\u 2 4u\u d + Bu\u\ 4u\u A + 12ulu 2 u 3 +4u x u\ Throughout the remainder of this paper, we let Figure 1 (1.2) R mn (u lr Ui,-) = R mn be the element in the m th row n th column of the convolution array. By mathematical induction, it can be shown that (1.3) R 1n = u?, (1.4) R 2 = nu n f 1 u 2, (1.5) R 3 = nut 1 u 3 + [ n 2)uT 2 u%. (1.6) R 4 = nu n f 1 u ( n 2 ) u1 2 u 2 u 3 + ( 3 ) u n f 3 u 3, (1.7) R 5n = nu n f 1 u 5+ ( J ) u"f 2 (u 2 3+2u 2 u 4 ) + 3 [ 3 ) u^u^s* ( " 4 ) u"f 4 ul (1.8) R 6n = nu n f 1 u 6 + 2( n 2) u n f 2 (u 2 u 5 + u 3 u 4 ) + 3 ( " 3 ) u n f 3 {ulu 4 + u 2 u ) + 4(" 4 )ur 4 u 3 2u 3 +(" 5 )ur 6 u 5 2, 193

2 194 GENERALIZED CONVOLUTION ARRAYS [OCT. (1.9) R 7n = nu n f 1 u 7 + i n 2 \ u n f 2 (ul + 2u 3 u 5 + 2u 2 u 6 ) +\ 3 ) u n f 3 (u% + 3u 2 u 5 + 6u 2 u 3 u 4 ) (1.10) + (" 4 ) u"f 4 (4u 3 u 4 + 6u 2 u 2 ) + 5 ( ) u f 5 u 4 2u 3 + ( ) u ^ i / f, R 8n = nu n f 1 u ( n 2 j u n f 2 (u 2 u 7 + u 3 u 6 + u 4 u 5 ) + 3 ( J ) u n f 3 (u%u 6 + 2u 2 u 3 u 5 + u 2 u% + u u 4 ) + 4 [ n 4) u n f 4 (u 3 2u 5 + 3u 2 2u 3 u 4 + u 2 u 3 3)+5 f ) u n f 5 (u 4 2u 4 +2u 3 2ul) + 6("J)ur 6 u 5 2u 3+ ( n 7)ur 7 u 7 2. The purpose of this article is to examine the general expression for R mn to find a formula for the generating function for any row of the convolution array. 2. PARTITIONS OF m AND R mn A partition of a nonnegative integer m is a representation of m as a sum of positive integers called parts of the partition. The function n(m) denotes the number of partitions of m. The partitions of the integers one through seven are given in Table 1. Table 1 Partitions of m -n(m) 1 2, ,1+2, , 2 + 2, 1 + 3, , , 2 + 3, 1 + 4, , , , , 3 + 3, 2 + 4, 1 + 5, , , , , , , , 1 + 6, 2 + 5, 3 + 4, , , , , , , , , , , Comparing the partitions of m, for m = 7 through m = 7, with the expressions for R mn it appears as if the following are true. 1. The number of terms in R mn is equal to ir(m The number of expressions whose coefficient is ( j ), for/= 7,2,, m - 7, is the number of partitions of m - 1 into / parts. * ' 3. The power of u t +i in an expression is the same as the number of times t occurs in the partition of m The numerical coefficient of an expression involving f n. J, for/^ 7, 2, 3, -, m - 7, is equal to the product of the factorials of the exponents of the terms of the sequence \ U n\n=1 in the expression divided into/factorial. The exponent for u x is not included in the product. In [4], it is shown that these are in fact true statements. That is, (2.1) where A77-7 Rmn(uu U 2, -) = J2 ( ) ut kp mk(ui, "2, ~'h (2.2) Pmk(Ui,u 2,u*,-) The first difference of a function f(x) is defined as k! a *u a *... a u m a 2!ch- f "'<>>m-fl u 2 u 3 u m ' k = OLj +a a n it() 3. SOME FINITE DIFFERENCES

3 1975] GENERALIZED CONVOLUTION ARRAYS 198 (3.1) Af(x) = f(x+1)-f(x). In an analogous fashion, we define recursively the n th difference A n f(x) of f(x) as (3.2) A n f(x) = A(A n ' 1 f(x». In [3], we find (3.3) Y. ( ~ 1>X \ x 1 fm = ) x=0 * Using mathematical induction, it is easy to show the following. (-V m ~ 1 & m ~ 1 f(0)- Theorem 3.1. \\f(x) = l r - * + s jthena'vm = (-1) n ( r ~ x j!_~ n \ Theorem 3.Z \if(x) = ( r + x } +s \ then A n f(x) = [ r JO.** ) - Applying (3.3), we then have Theorem 3J. If fix) = ( r ' xi ' s ) then /fix { E " f m - 7 \ / r- \ x ) \ x + s\ 1 Jr + s m + 1\ -[ l-m+1 ), x=0 Theorem 3A. tf f(x)= [ r + * +s \ then x=0 4. THE SVIAIN THEOREM Combining (2.1) with Theorem 3.3., we see that, whenever^ = 1, we then have ( n-k+1\ ^ ^mj E ^ ( m ; ' ) v ^ - E '-"* ( V ) ("T? ) p» k'~0 k=0 M E % E <-»* ( m * f )("T') - L *w( ; - n )- *w-» Now, the only way to partition m - 1 into m - 1 parts is to let every part of the partition equal one. Hence, by (2.2), we have Pm, = u 2 so that (4.1) E '-"*( V )»«K**H 'IF'- k=0 From (4.1), it is easy to see that the generating function g m (x) for the sequence R m,n+i\ ^=Q > where*// = 1, is of the form (4.2) gmm = JlmL x L=Y f R m,n+1* n (1-x) m n=0 In order to determine the generating function g m (x) for the m th row of the convolution array, it is necessary to determine what is commonly called "Pascal's attic." That is, we need to know the values for the columns corresponding to the negative integers zero subject to the condition of (4.1). With this in mind, we develop the next two theorems. oo

4 106 GENERALIZED CONVOLUTION ARRAYS [OCT. Th0awn4.l If m > 2 u x = 1 then R mf0 = 0. Proof. Letting/7 =/7? -2\n (4.1), we have m-2 =0 Ar=7 /n-/ + U _ \~^ / 1 im+k I m - 1 \ p,,. I/JT' 2 = - 2^ (~ 1) [ k ) R mk +u 2 *«/ By (2.1), using/as the variable of summation, Theorem 3.4 with r = s = 0, we obtain j=1 - (-D m E *w E '-"* ( m,v)( J ) +u? 1 M 1=1 k=0 Z-f r J \ j-m the theorem is proved. Theorem4.2. If n > l,m >2m&u 1 = 1 then + 1 j 0 Proof. We shall use the strong form of mathematical induction. Replacing n by m - 3 in (4.1) following the argument of Theorem 4.1 where we let r = 0 s = - 1 in Theorem 3.4, we have m-2 {-ir'rm.-l = E ^k+1 [ m V) R,m-k-2+ur 1 - E <- 1 > m+k [ m ~k 1 Wjk-^2 k=0 Recalling that = (-rr E % E <-" k (V){ k 7? ) j=1 r mk + "2~ 1.. = <-D m E >w E '-"* ( m k 1 ) ( * 7 ' ) - ( ~ 1>m E 'w ( 7 H r 7 /=? fc-0 ' / = / /77- f /77-1 EM/-^/W- / r EM7K~'. 1=1 1=1 if/? > 1, /7? > O f _ ^ ) = $for all /7 provided m > 1, we have so that {-D^Rm^ = ~P m,-(-1) m (-1) J Pmj + u$~ 1

5 19751 GENERALIZED CONVOLUTION ARRAYS 197 R m,-1 = E (-V'PmJ 1=1 the theorem is true for/7 = 1. We now assume that the theorem is true for al! positive integers less than or equal to t Replacing n by m - t-3 in (4.1), we see that m-2 f - / r V / ) = E <- 1 > k+1 ( m ~ k 1 ) R > - t - k ~ 2+u 2 k=0 where the last equation is obtained by the induction hypothesis. Multiplying by (-1) m ~ 1 introducing k = 0, one has 7=7 =0 '' /=* M r %, x ; (-D k ( V ) ( '-**') ^ r ' 7=7 /f=/ m ~ 1. x = ^ / _, M 7 - ^ 7 )* '-«/ (7 / )'W +^^', = M ; ' ( < ; ' ) / W, /=/ where the second equation is obtained by use of Theorem 3.3 with r= t s = j the theorem is proved. We are now in a position to calculate the generating function for the /?? t/7 row of a convolution array when*// = 1. When m = 1, we see that /?/ //7 = 1 for all n > 0 so that < 4-3> 91M = E *" = 7T7 ' oo m-7 By (4.1), we have R,n+i-E '-"**' ( V J w ^ / ^ r ' so that when m > 2, we can use (4.2) to obtain <* A:=7 /7=0 fc=7 /?=$ oo,,\ m-l\ m-i / /c-/ /r-7 \ it m-7 M-- M^(VKU^ E / W W ^ ^ j - M^'(» ;') W^w + ^- n x-"-' U ^ -. n=0 / \ n=1

6 198 GENERALIZED CONVOLUTION ARRAYS OCT Hence, k-1 (4.4) g m (x) = JzLJEl / m > 2. (1-x) m For special sequences with u-i = 1, the polynomial in the numerator of g m (x), m > 1, is predictable from the convolution array of the sequence. This matter will be covered by the authors in another paper which will appear in the very near future. REFERENCES 1. V.E. Hoggatt, Jr., Marjorie Bicknell, "Convolution Triangles for Generalized Fibonacci Numbers," The Fibonacci Quarterly, Vol. 8, No. 2 (April 1970), pp V.E. Hoggatt, Jr., Marjorie Bicknell, "Convolution Triangles," The Fibonacci Quarterly, Vol. 10, No. 6 (December 1972), pp Charles Jordan, Calculus of Finite Differences, Chelsea Publishing Co., 1947, pp John Riordan, Combinatorial Identities, John Wiley Sons, Inc., 1968, pp ******* LETTER TO THE EDITOR February 20,1975 Dear Mr. Hoggatt: I'm afraid there was an error in the February issue of The Fibonacci Quarterly. Mr. Shallit's proof that phi is irrational is correct up to the point where he claims that 1/0 can't be an integer. He has no basis for making that claim, as 0 was defined as a rational number, not an integer. The proof can, however, be salvaged after the point where p is shown to equal 1. Going back to the equation p 2 - pq = q 2, we can add pq to each side, factor out a q from the right: p 2 = q(q + p). Using analysis similar to Mr. Shallit's, we find that q must also equal 1. Therefore, (p-p/q = 1/1 - /. However, / = - / / ^ 7 ; thus, our assumption was false, 0 is irrational. Sincerely, s/david Ross, Student, Swarthmore College *******