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Journal of the Egyptian Mathematical Society (2015) 23, 297302 Egyptian Mathematical Society Journal of the Egyptian Mathematical Society /locate/joems ORIGINAL ARTICLE On an explicit formula for inverse of triangular matrices Pinakadhar Baliarsingh a,*, Salila Dutta b a Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India b Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar, India Received 22 April 2014; revised 26 May 2014; accepted 1 June 2014 Available online 5 July 2014 KEYWORDS Difference operators BLam Cesaro mean operator; Riesz mean and generalized mean operator Abstract In the present article, we define difference operators BLam m 2 N0 f0; 1; 2; 3; . . .g, where amP a1; . . . amg, the set of P k k2N 0 66 convergent sequences a iai0imof real numbers. Indeed, under different limiting conditions, both the operators unify most of the difference operators defined by various triangles such as D; D1; Dm; Dmm 2 N0; Da; D aa 2 R; Br; s; Br; s; t; Br; s; t; u, and many others. Also, we derive an alternative method for finding the inverse of infinite matrices BLam 47A10; 46A45 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. 1. Introduction, preliminaries and definitions Difference operators are one of the important subclasses of Toeplitz operators where most of them are reduced to triangles under different limiting conditions. Triangular matrices have several applications in scientific computations and engineering, and the most useful contributions are solving the system of * Corresponding author. Tel./fax: +91 9090027609. E-mail addresses: (P. Baliarsingh), saliladutta516 (S. Dutta). Peer review under responsibility of Egyptian Mathematical Society. Production and hosting by Elsevier linear equations and finding spectral properties of bounded linear operators. Several methods have been employed to find the inverse of a triangle such as back ward substitution and elim- ination methods. The main idea of this note is to study certain triangles and derive an alternative method for their inverses. Let w be the space allreal valued sequences and for m 2 N0; am s sk, and t tk be three sequences in U and Xn Tn :tkn 2 N0: 0a01 a00 B. B. Ba02 B B B Lnk B. B0 B B a0m B. B. B. 00. . .00 a100. . .00 a11a20. . .00 . . . a1m 1 a2m 2 . . . am00 a1ma2m 1 . . . am1 am10 . . . . . . 1 . . . CC C . . . CC . C .C .C; C . . . CC C . . . CA . . . k0 Then the Cesaro mean of order one and Riesz mean with respect to the sequence t tk are defined by the matrices C1 cnk and Rt rtnk, respectively (see 9,10), where 1 cn1; 0 6 k 6 nn kN nk :0;k n; ;20 tn tT ;0 6 k 6 n rnk: 0n; k n ; n; k 2 N0: and a00 00 B0 B . B .Unk B . B B 0 B BB 0 B B B B . a01 a02 . . .a0m0 a10 a11 . . .a1m 1a1m 0a20 . . . a2m 2 a2m 1 . . . 00. . .am0am1 00. . .0am10 . . . . . . 1 . . . CC C . . . CC . C .C .C: C . . . CC C . . . CA . . . The generalized mean of the sequence x xk can be com- puted by using Ar; s; t transform ofx (see 11), where Ar; s; t represents an infinite matrix ank and sn k tk ;0 6 k 6 n ank : 0;rn k n ;n; k 2 N0 : 2. Main results In this section, we study certain results concerning the linear-ity, boundedness, and inverse properties of the infinite differ- Different kinds of triangles via difference operators have been studied by various authors. For instance, triangles such as double banded D, triple banded Br; s; t, fourth banded Br; s; t; u, and m 1 banded Dm matrices have been intro- duced by Kzmaz 1, Furkan et al. 2, Dutta and Baliarsingh 3 and Et and Colak 4, respectively. Altay and Basar 5 and Dutta and Baliarsingh 6 have studied the spectral properties of difference operators Br; s and D2, respectively. In fact, the detailed study of these operators involving topological proper- ties, duals, matrix transformations and spectral properties is only possible by determining their inverse operators. Recently, Baliarsingh 7 and Dutta and Baliarsingh 8 have introduced fractional order difference matrix Da and m 1sequential band matrix Bam06i6m Theorem 2. 8, Theorem 2If ak00 for all k 2 N0, then an explicit formula for inverse of the difference operator BLamk n aj 0 Dn k am06 k 6 n 1 ;n; kN0: j k 2 Q; 0kn where : ak2 ak1 . k0 Dn a m ak m n P 1 : . a nk 11 inversion of triangular matrices299 1 Proof. We apply induction method for proving this theorem. an 0 ;k n 2 k aj 0 Dk n am0 6 k 6 n 1 ;n; kN0: j n 2 x0y0 Q; 0kn a00 where: y1a01y0 x 1 a10a00a10 Dnnam a3 0a20 a3 0 a1 0 a2 0 a3 0a10 a3 0 x1 1 ;order n 1 for all n 0; 1; 2; . . On solving the system of 1Q0 xnyn ; y3 D1 y2 D2 y1 D3 y0 jn 1 x3a30 j32aj0j31aj0 j3 0aj0 yn 2 an 2 1 yn 1 Q xn 2 an 2 0 an 2 0 an 1 0 an 0 nTheorem 2 holds, and by assuming Therefore, forQ1; 2; 3, QQ 6 n 1n 2n 3 an 20an 1 0 an 0an 20 an 0 yn a1a1a2 n 1n 2n 2 so for all k n, and we have D y n1 nn yn 1D2D3yn 2yn 3yn 2 D1 yn 1D2 yn xn a 0 na0na0na0n 1n; n jn 1 j jn 2jjn 3 j an 2 0 n 2aj 0j n 2aj0 Q nDn0y0QQ jn n n . . .1yD y QD y D y n a 0 xn 3 Q1n 22n 13n . Q Q j0j n 3 a0 n 2n 1na0 k nQ n 3 jn 3aj0 jn 3aj0jn 3j Now, for1, we may deduce.Q . 1ynD1n 1yn 1 xn 1y an1 n . . . an10 “n1 an0 jn 1aj0 For deducing x0, we apply induction method as discussed in 0Qn 2proof of Theorem 2.h D yyDyn2 nn0n 11 1jn 0aj0 ! a n 12an 10n 1 aj 0Corollary 1. The inverse of generalized fractional difference Q 0jn 2 operator Da; aR is given by D y Qy 2 00 1n 1n 1an1 . . . n 1aj0 ! . . . 0 a00# Qj0 n 1 a 1; n k 1 a ik n yn1an1ynan1 D1 an 1 2 an0D 8i0 ;n; k N0 : n 1 yn 1 nk n k! ; 0 6 k 6 n 12 an 1 0 an 0 an 10aj 0 j n 1! n an1 D2an 12 an0D1an23 an10an0 y:a n1 aj 0!n 2Proof. The generalized fractional difference operator Drepre- 0 Qjn 2 Dny nsents a lower triangular Toeplitz matrix (see 7 for detail), i.e. n1Dn1 yn1 1 . . . 1n1aj0an10n1aj01000 0 . . . n 1 Q jn 2 Q 0 a 100 0 . . . 0jn D 2 y n 1 D 3yn 2 n 1Dn 1y001 a a 1 jn 1aj 0 jn 2aj 0 j0 aj 0 B 2! a10 0 . . . C n1n1. . .1n1 QQQB3! 2 !a1 0 . . .C Da Ba a1a 2 a a 1 C: BC This completes the proof.hB C Baa 1a 2a 3a a 1a 2aa 1C B 4 ! 3! 2 !a 1 . . .C 2 BC Theorem 3. If ak00 for all kN0, then an explicit formulaB C k n Dnka81n kDnkkDa;0 6 k 6 n1;n; kN0; Tn ; kn 2 1 tn n k2 Rnk8 t ;kn1 ;n; k N 0; nt Tn k 1 : where : 0;otherwise DnkDa Dn0Da a100 . . .0 a a 1 2!a10 . . . 0 a1 . . .0 a a1a 2a a1 3 !2! 4!3!2! a . . .0 a a1a2a 3a a1a2a a1 : . . 0a 1aa .a 1aa.a 1aa .a . . . . . . n1n1n11n2n21n3 n!n1!n2 !a However, using suitable row or column operations, the determinant Dn D can be reduced to 1100 . . .0 a 1 2 !a10 . . . 0 a1 . . .0 a1a 2a a1 3 !2! 4!3!2! a . . .0 aa1a2a 3a a1a2a a1 . . a a1 1 a .a 1aa .a 1aa .a . . . . . . n 1 n1n11n2 n21n3 n!n1 !n2 !a 2 1 000 . . .0 a 1 2 !110 . . .0 a1 . . .0 a1a22a1 3 !3 a . . .0 a1a2a 3a1a2a a1 4 !42! . . 1a .a 1a .a 1aa .a . . . . . . n1 n 1n11n2 2n1n21n3 n!n!n 2 !a aa 1 a 2 23 100 a 1 2! 10 1221 a a a 1 3!3 a 1a 2a3a1a23a1 4!44 . n a 1.a n1n 1a 1.a n22n 1n 2a 1.a n33n 1n 2 1n!1n! 1 n! . . .1 a a1a2a3an1 n 234n 1naa 1a 3a 4 . . . a n 1: n! 0 . . .0 0 . . .0 1 . . .0 a . . .0 . . . a Therefore, it is being concluded that 2 : 0;k n Proof. The proof of this Corollary is analo