ONE EXPRESSION FOR THE SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS JERZY POPENDA

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1 proceedngs of the amercan mathematcal socety Volume 100, Number 1, May 1987 ONE EXPRESSION FOR THE SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS JERZY POPENDA ABSTRACT. Explct formulas for the general solutons of homogeneous and nonhomogeneous second order lnear dfference equatons wth arbtrarly varyng coeffcents are obtaned. 1. Introducton. In ths note we gve some explct formulas for the solutons of second order dfference equatons. The obtaned expressons can be used both n theoretcal nvestgatons and numercal methods. Note that smlar results are not known for dfferental equatons. To obtan these formulas we shall ntroduce some specal sum operators. We shall use the followng notaton. (a) N; the set of postve ntegers, (b) R; the set of real numbers, and (c) xn; the value of functon x at the pont n G N. Letx.N^R. We defne 1 n () Y(nk)X>:= Y Uxkk forn>l,fc>0, d,...,d =0 =1 and (Ü) We assume ran dd + =0 =l,...,n l Y(1<k)X'- Y *?+* ( 1=0 Ex = 1 V* x 1 Y^ x = 0 (0,fc) l ' ^(-l,fc) % ' -f(-2,k) * k-t (v) Y x = > T x = 1 for a11 k,tg N, and =k k-t =k (v) 0 = 1, 01 = 0. The forward dfference operator s defned as usual; Axn = xn+1 - x, A2xn = A(Axn). Receved by the edtors November 12, Mathematcs Subject Classfcaton (1985 Revson). Prmary 39A05. Key words and phrases. Fnte dfference equaton, general soluton Amercan Mathematcal Socety /87 $ $.25 per page

2 88 JERZY POPENDA 2. Man results. THEOREM 1. Let q: N -* R. Then every soluton of (1) xn+2 = xn+ +qnxn, ngn, can be represented by the formula (2) xn = qx E(n_4>2) 9 + ** Y{n-s,) *' U * 2' where x,x2 are arbtrary real constants. PROOF. The proof of ths theorem follows by nducton. We see, by (), that (2) becomes an dentty for n = 2. Furthermore t s not dffcult to check (2) for n = 3,4,5 and compare 13,2:4, 5 wth the same x calculated n successon from (1). Let (2) hold for n = m and n = m + 1. We prove that t remans true for n m + 2. From (1) we get Xm+2 = Xm-\-l ~r ÇrnXm =^ (m_3,2)*+a*e(m_2)1)* + Qrr, Y ^(m_4,2)* + a;2e(rr_3,1)* = qx ím_3,2)*+*» '(m-4,2) + X2 (m-2,l)9î + 9mX,(m-3,l) = qx Qm-2+2 Y{m-3,2) Q + 9m m (m_4,2) Ql + x2 9m-l + l 2-,(m_2)l) * + 9m-l+l9m-2+l Y{m-3,l) * * '(m 2,2) We shall now consder the equaton '(m 1,1) (3) xn+2 = axn+ + qnxn, ngn, where q: N R, a s any fxed real constant. We know that for the case a = 0 the soluton of (3) s Now let a/0. Xn = < k xttq2j- for n 2fc + 1, fc = 0,1,..., J=l fc forn = 2fc + 2, fc = 0,1,... After dvdng (3) by an+2, t can be put nto the form yn+2 = yn+ + -ôî/n,

3 SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS 89 where yn = xn/an, ngn. Hence by Theorem 1 we obtan where pn = a 2qn, n G AT. Therefore -2 1 Y, a n, P«+ ^ *»(n 4,2) ^(n-3,l)pl' (4) Xr = an^q^yn-,2)pl+an^x2^(n-o,x)pl' " " 2' wth p defned above and X,x2 arbtrary constants. In explct form expresson (4) s equvalent to Cn = XQ Y d,...,dn-4=ö dd+ =0 l,...,n E d,...,d _3=0 n »= n 3 IR+ L=l (j-'-s-îe":,1*) (-2y"n 3d.) ^ r a^ -n= >J, n > 5. =l,...,n 4 Equaton (3) s connected wth (5) A2x = r x, ngn, because (5) can be wrtten n the form the soluton (4) of (3), we get the general soluton of (5). Specf- Havng obtaned cally, x +2 = 2xn+ + (l + r )xn, ngn. xn = 2-3(l + n)x Y(n_Aa) «+ 2 X2 Y{n_3A) «.» > 2, where zn = j(l + rn), ngn, and,x2 are arbtrary. We am our attenton to the equaton (6) Xn+2 = anxn+r + qnxn, ngn, where o: N > î\{0}, q: N -> R. We shall set bn+2 = an, n G N, b = b2 = l. Dvdng (6) by YY bj and puttng we have 7/n X, ïïh 2/n+2 = J/n+l yn, 0n + 2 0n+l Consequently by Theorem 1 "*"> J=l 71 6 V. where ^ = ^E(n_4,2)^2 ( -3,l)^ Qn bn+2bn+l ngn.

4 90 JERZY POPENDA Therefore the general soluton of (6) s xn = xq f nj=2 EZ (n-4,2) + x2 J = l E( _3,l)2î' 71 > 2, where zn = qn/anan-, n > 2, z = q/a, and x,x2 are arbtrary constants. Havng solved (6) we can obtan the general soluton of a lnear homogeneous second order equaton (7) anxn+2 + bnxn+ +cnxn = 0, ngn, where a,b: N >?\{0}, and c: N >?. Ths soluton s (8) X = (-l)nx! ("l)nx2 Ol f j = 2 3, n* where x,x2 are arbtrary constants and J -'(n-4, 2) Z CnQ*n 1 Zn = for n > 1. On&n-l EZ, 71 > 2, (n-3,1) " Consderng the lnear nonhomogeneous equaton we have the followng result. THEOREM 2. Letb,c: N -> Z\{0}, r: N ^ R, (9) 5^(n) * # 0 /or n g N, where (10) Then the general soluton of (H) s Xn = (-1)"X2 cn OnDn-l n > 2. n+2 +bnxn+ + cnxn = rn, ngn, =1 + (-r+xlcl + (-!)" = -^(n-3,l)zl ^-'(n-3,1) -^(n-3,1) -, -^ J fc=l /or n > 2, where x,x2 are arbtrary constants, and fc+1 t -' n fc=l J=2 fc+1, n =2 J fc 5ZrJ%+1 yn = (-)r, '(n-3,1 (n-3,1) 71 > 2.

5 SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS 91 PROOF. In order to obtan (12), we consder the equaton (13) Cn+yn+2 + bnyn+ +yn = 0, ngn. The general soluton of (13) s, by (8), f (14)» = (-1)" ^yy n3=1 cj+ h (n-4,2) Z + V2 > Z, 1 y -^(n-3,1) * where zn s defned by (10). Let yn satsfy the ntal condtons a result, we get from (14) y = 0,y2 = 1. As (15) yn = (-)1 n Cj + l Y 71 > 2. '(n-3,1 (n-3,1) Under the hypothess (9), the soluton (15) vanshes at no pont n, n > 2. On multplyng (11) by yn+ and wrtng (13) n the form bnyn+ = -y - Cn + yn+2 we have (yn+lxn+2 ~ Cn+yn+2Xn+l) - (VnXn+1 -CnVn+lXn) = rnyn+. On summng the above equaton wth respect to t one gets (16) yn+xn+2 - cn+yn+2xn+ = y\x2 - cxy2x +^r-yj-+, nejv. Settng y = 0 and y2 = 1, dvdng (16) by yn+yn+2 XY Cj+, and summng next wth respect to n, we have n 3=1 Xn+2 Vn + 2 n 3=1 Cj+l fc=l X2 CX Y 7 = C3 + l yk+\yk+2 k Thus Zn+2 = x2yn+2 yn+2 Y\ Cj+ - CXyn+2 3 = 1 n cj+] fc= fc= 3=1 Cj+l n cj+] = Efc= fc+ t t fc n -Y J=l Cj fc+, 3=1 3 C 111., 111., n ' J J yk+yk+2 rw+> ngn. Hence, by (15) we nfer that (12) holds for n > 2. Furthermore, by () and (v), (12) s also vald for t = 2. Q.E.D. Note that equaton (11) can possbly be solved even n the case when (9) s not fulflled, that s, even f the yn gven by (15) vanshed for some n. The yn gven by (15) s one partcular soluton of (13). The general soluton s gven by (14). If the yn gven by (15) vanshed for some t > 2, then any other partcular soluton

6 92 JERZY POPENDA obtaned from (14) whch dd not become zero for any n G N could be used. Ths soluton rather than (15) should be substtuted n the followng equalty: Xn+2 = yn+2 n+1 nj'=2 X2 y2 fc=l 1 yk+yk+2 fc+, 3 = 2 3 yxx2 - cy2x + Y r%+l 3=1 71 G TV. However, the obtaned expresson for the soluton of (11) wll be more complcated n any case. Fnally, we see that the equaton (17) anxn+2 + 6nxn++c xn = r, where a,b,c: N > R\{0}, r: N > R, can be solved by applyng Theorem 2. On dvdng (17) by an, we get. "n. Cn Tn Xn+2 H-Xn+ H-Xn = - whch s of the type consdered n Theorem 2. Therefore, to get ts soluton we need a nonvanshng soluton of the equaton Cn on -yn+2 h y«+ + yn = 0. On+l an If for nstance ^\ ^ Z ^ 0 for all n G N, where zn = cnan-/bnbn-, we can wrte the general soluton of (17) n the form Xn = ('IT n3=1 bj Y '(n-3,1 Z X2 - + E fc= C s~^ Xl 2^ a fc = l fc+ nj = 2 fc + 1 n J=2 1 k EÍ ~[ aj yj+ Tl> 2, where x,x2 are arbtrary constants, zn s defned above, and yn = (-y an-l f,c +1 EZ, n > 2, y = 0. (n-3,1) ' - ' y 3. Nonlnear examples. Varous equatons can be made lnear by a change of varable. In partcular, such a type s Rccat's equaton (18) anxn+lxn +bnxn +Cn = 0, where a,b,c: N > 2\{0}. Make the change of varable xn yn+/yn, wth yn ^ 0 for all n G TV. Then (18) becomes a yn+2 + bnyn+ + cnyn = 0.

7 Hence, by (8) SOLUTIONS OF SECOND ORDER DIFFERENCE EQUATIONS 93 bn-l (ClE(n-3,2)^+bl:ElE(,l)^) n>2 a-n-1 [ ClE(n-4,2)2* + 6lXlE(n-3,l)^ j ' where zn = c an_/6n6n_, n > 1, x s an arbtrary constant. For yn ^ 0 we need c >^ Z + 61X1 > Z ± 0 for all n G N. -^(n-3,2) -^(,1) The second example treats the equaton (19) anx +x + 6 x + + cn = 0. Here, by puttng xn = yn/yn+, yn ^ 0 on TV, we get Supposng a,b,c: N > Ä\{0}, and cnyn+2 + onyn+ + anyn = 0. o-x > > z ^ 0 for all ng N, -^(n-4,2) * * -^(n-3,1) ^ ' where 2n = ancn_/6n6n_, n > 1, we obtan, by (8), the expresson of the soluton of the consdered equaton. That s, the soluton x of (19) s gven by Cn-l Í alxle(n-4,2)^ + 6lE(n-3,l)Z' ] n>2 n-l \ alzle(n-3,2)^ + 6lE(,l)^ j ' 4. Note that the operators ntroduced n ths paper can be used wth some modfcaton for recurrence equatons of hgher order. Insttute of Mathematcs, Techncal Unversty, Pozan, Poland