The main objective of this paper is to propose the matrix representation of umbral composition and investigate the effect of delta operator on umbral composition by using the sequential representation of delta operator in finite operator calculus

Elango, C

Maheswaran, A

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I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6392 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m   Effect of Delta Operator on Umbral Composition in Finite Operator calculus A Maheswaran1*  ,  C.  Elango, 2 1 , 2 Department of Mathematics, Cardamom Planters’ Association College, Tamilnadu, India. maheswarancpa@gmail.com ABSTRACT  The  main objective of this paper is to propose the matrix representation of umbral composition and investigate the effect of delta operator on umbral composition by using the sequential representation of delta operator in finite operator calculus. Keywords:  Delta operator; Basic polynomial sequences; Sheffer sequences; Umbral Operator; Umbral Composition. Academic Discipline And Sub-Disciplines Mathematics, Umbral Calculus.  SUBJECT  CLASSIFICATION  Umbral Calculus 1 . INTRODUCTION  The theory of umbral calculus studies the analogies between several sequences of  polynomials pn and the monomials  {xn/n∈ ℤ+⋃{0}}. It is developed in 1970’s by G.C.Rota and he linked umbral composition with Finite Operator Calculus.  Currently, Umbral Calculus refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appel sequences. Emil.C. Popa [3] presents the Taylor formula in Umbral Calculus. Some applications of the Umbral Calculus are discussed in Di.Bucchianico [1].  G.C.Rota [9] serves us a grips introduction and a guide to the growing literature on combinatorics and it contains a detailed study of umbral operator and umbral composition.  The goal of Ira M.Gessel [4] is to show,how the classical umbral calculus can be used to proving some interesting formulas not all can easily proved by other methods.  Roman  M.Steven and G.C.Rota [8] is a detailed study of umbral calculus and applications of  combinatorics. A.Maheswaran [6] introduced a new approach to the delta operator as a sequence of real numbers.  The characterization of Sheffer umbrace is analyzed in [2]. The aim of the present paper is to propose some results tied to the umbral operator and umbral composition in Umbral Calculus.  The rest of the paper is organized in four sections.  In the second section, we introduce some basic definitions, operation, notation and theorems from G.C.Rota we need.  In the third section, we discussed about the sequential representation of delta operator. Due to the discussion on umbral operator and umbral composition, we propose the matrix representation of umbral composition in fourth section. Finally the effect of delta operator on umbral composition of two basic polynomial sequences are analysed in fifth section. 2 .  PRELIMINARIES  Let  p(x) be a polynomial in one variable defined  over a field F of characteristic zero and let   {Pn(x) / n ∈ ℤ+ ⋃ {0} }, be  a sequence of polynomials, where pn(x) is exactly of  degree n. Definition 1. i. An operator  Ea  is said to be a shift operator if  Ea p(x) = p (x+a), for all polynominals p(x) in one variable and for all real ‘a’ in the field. ii. A linear operator T which commutes with all shift  operators is called a shift invariant.  i.e .  T E a  =  E a T. iii. A shift invariant operator Q satisfying that Q x  is an non zero constant is called a delta operator. Definition  2. Let Q be the delta operator, A polynomial sequence pn(x) is called sequence  of basic polynomials for Q if : i. P0(x) = 1 ii. Pn(0) = 0, whenever n > 0 iii. Qpn(x) = n pn-1(x) A trivial example for basic polynomial sequence is {xn}. Theorem 1. i. Every delta operator has a unique sequence of  basic polynomials. ii. If Q is a delta operator, the Q a = 0,  for every constant ‘a’. I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6393 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m   iii. If p(x) is a polynomial of degree n, then Qp(x) is a polynomial of degree n-1. Proof.  Refer to G-C Rota [3] By above [i],[ii] and[iii] in Theorem 1, Delta operator Q possesses many of the properties of usual derivative operator D. Definition  3. A polynomial sequence sn(x) is called a Sheffer set or a set of Sheffer polynomials for the delta operator Q if i. s0(x) = c ≠ 0, ii. Qsn(x) = n sn-1(x) For example, The monomials {xn: n = 0,1,2,3,….},  the Abel polynomials, the Hermite polynomials, the laguerre polynomials, the Difference polynomials, the Euler polynomials, the Mott polynomials, the Bell polynomials, the Swiss-Knife polynomials are very popular Sheffer polynomials. 3 . SEQUENTIAL REPRESENTATION OF DELTA OPERATOR  Q Now we attempt to formulate the delta operator interms of sequence of real numbers.  By Theorem (1) and the definition  for basic polynomial sequence, we obtain the following Theorem. Theorem 2. For the monomial {xn :  n ∈ ℤ+ ⋃ {0} }, and for each 𝛼r  ( r = 1, 2, 3, . . .n)  an arbitrary real numbers , Q( xn ) =  𝑛𝑟  αr 𝑛𝑟=1xn−r        (1)  Proof. Taking Q(x) =  𝛼1 ≠  0 and construct Q(x2) = c0 x + c1.  Since Q is shift  invariant, we have Ea Q( x 2 ) = Q Ea ( x 2 .  Solving we get c0 = 2 𝛼1 and c1 is a new independent constant which may be taken as 𝛼2. .  Hence Q ( x 2) = 2 𝛼1 x + 𝛼2.  Thus the theorem is true for n=1 and 2. Let us assume that the result is true for all n = k. Therefore, Q(xk) =  k𝑟  𝛼 𝑟   𝑥𝑘−𝑟k𝑟=1 =  k1  𝛼 1 x k-1 +  k2  𝛼 2 x k-2 +…+  kr  𝛼 r x k-r +…+ 𝛼k   (2) Since{xn} is a basic polynomial sequence, it satisfies Q pn (x) = n pn-1(x) and hence we have, Q(xk)=k xk-1                  ( 3 ) From (3), we see that the delta operator Q is a usual derivative D.                                                                              From (2) and (3),  k1  𝛼 1 x k -1 +  k2  𝛼 2 x k - 2 +…+  kr  𝛼 r  x k - r +…+ 𝛼k   = k x k -1                                       (4) By comparing the corresponding terms, we have   𝛼1 = 1 and 𝛼 j = 0 ,  j = 2,3,…k Therefore, the result is true for n = k means that                                                        𝛼 1 = 1 and 𝛼j = 0  ( j = 2,3,…k)     (5) Now we have to show that this result is true for n = k + 1 Q(xk+1 )= Q(x k  x) = Q(x k) x + Q(x) xk  = (k+1) xk Thus we have                                         Q(xk+1 ) =  (k+1) x k                        (6) On other hand, using the property that  Q pn(x) = n pn-1(x),  we have Q (xk+1) = (k+1)  pk (x) =(k+1) xk     (7) From the Equations (6) and (7), we conclude that the result is true for all n = k+1                                                     Thus we proved the Theorem (2).  The following Table (1) contains first few polynomials Q (xn), for each degree n. I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6394 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m   n Q (xn) 1 α1 2 2 α1 x + α2 3 3 α1 x2 + 3 α2 x +α3 4 4 α1 x3 + 6 α2 x2 + 4 α3 x + α3 5 5 α1 x4 + 10 α2 x3 + 10 α3 x2 + 5 α4 x + α5 6 6 α1 x5 + 15 α2 x4 + 20 α3 x3 + 15 α4 x2 + 6 α5 x + α6 Here, Q(xn) has n independent parameters, i, (i = 1,2,3,….n).   These parameters are unique.   The characterization of delta operator is determined by the values of ’is  (i = 1,2,3,…n).  To study the delta operator Q, we need analyse only this sequence (1) Theorem (2).  This sequential representation of delta operator plays a vital role in deriving many interesting results for basic polynomial sequences and seffer sequences.  In next section, we study more about umbral operator and umbral composition. 4.  MATRIX REPRESENTATION FOR UMBRAL COMPOSITION  The study of umbral composition starts from a classical operator namely ‘umbral operator’. Definition 4. An operator T is said to be umbral operator if T maps some basic polynomials pn(x) into another basic polynomials sequence qn(x). In another words, T is umbral operator if Tpn(x) = qn(x), where pn(x) and qn(x) are basic polynomials sequences. Note that an umbral operator is not shift invariant. Definition  5.  The umbral composition of two polynomial sequences an(x)  = nk 0ank  xk  and bn(x) is the sequence of polynomials  cn(x) defined by                                                  c n (x) = nk 0ank  bk(x)                                                                                   (8) The notation for umbral composition is given by                                                   cn (x) = an ( b (x) ) The umbral composition is evidently associative but not commutative.  Infact, simple polynomial sequence form a group under umbral composition, the identity element being  {xn}.  The following results are connected with umbral operator, delta operator and basic polynomials sequence. Theorem 3. Let T be an umbral operator, then 1. T-1 exists. 2. If Q is a delta operator, then TQT-1 is also a delta operator. 3. T maps every sequence of basic polynomials into a sequence of basic polynomials. Proof. Refer to G.C.Rota [9].  The next result determines the operators corresponding to the umbral composition. Theorem 4. Let sn(x) and tn(x) be Sheffer sets relative to the delta operator Q and P, and to the invertible shift-invatiant operator S and T, respectively. Let qn(x) and pn(x) be the basic sets for Q and P, and let the indicators of S, Q and P be  S = s(D), Q = q(D) and P = p(D) where s(t), q(t) and p(t) are formal power series.  Define rn(x) to be the umbral composition of  sn(x) and tn(x), in symbol, rn(x) = sn ( t (x) ),  then rn(x) is a Sheffer set relative to the shift-invariant operator T s (P) = t (d) s ( p(D) ) I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6395 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m   and the delta operator q ( p(D) ), having as basic set the sequence qn ( p(x) ). Proof.  See [9] An umbral interpretation of functional composition of formal power series is discussed in [5].  Now we discuss the general form of umbral composition in matrix form. Let an(x) = nk 0ank  xk and  bn(x) =nk 0bnk  xk Then by equation (8), the umbral composition of an(x) and bn(x) is given by : cn (x) = nokank  bkk xk In other words, c0 (x) = an0 b00 x0 + 0 x1 + 0 x2 + 0 x3 +  + 0 xn c1 (x) = a10 b00 x0 + a11 b11 x1 + 0 x2 +  + 0 xn c2 (x) = a20 b00 x0 + a21 b11 x1 + a22 b22 x2 ++ 0 xn   cn (x) = an0 b00 x0 + an1 b11 x1 + an2 b22 x2 +  + ann bnn xn Hence the general form of umbral composition can be written in matrix form as, )()()()()(3210xcxcxcxcxcn = 00 0010 00 11 1120 00 21 11 22 2230 00 31 11 32 22 33 330 00 1 11 2 22 33 330 0 0 0 00 0 0 00 0 00 00n n n nn nna ba b a ba b a b a ba b a b a b a ba b a b a b a b a b                 nxxxxx3210 Thus the general form of umbral composition is an (n+1) X (n+1) lower triangular matrix, whose diagonal elements are not equal to zero and it can be expressed as c k (x) =   nok A k j x k Where                                                                                           a k j  bk k      for k = j                                                     A k j =         a k j bj j      for k > j                                                                           0            for k < j                                                                                                                                        5.  EFFECT OF DELTA OPERATOR ON UMBRAL COMPOSITION  If a n (x) is a polynomial sequence, then there is a unique linear operator L such that  L( x n ) = a n (x), we say that L is the umbral representation of the sequence a n (x). Let b n (x) = xn  ,   n ∈  ℤ+𝑈 {0}. It satisfies b 0 (x) = 1 and b n (0) = 0, whenever n > 0. Hence it is a basic set and also Sheffer.  I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6396 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m   The first few polynomials are b1 (x) = x , b2 (X) = x2,  b3 (X) = x3,  b4 (x) = x4 and so on.  Fig 1:  Monomials b n (x) = { x n /  n ∈  ℤ+𝑼 {𝟎}  } From the property Q bn (x) = n bn-1 (x) and Table (1), we obtain the following proposition. Proposition 1.                                                                                                                              For the Monomials b n (x) = { x n /  n ∈  ℤ+𝑈 {0}  } ,  the characterization of delta operator being α1=1, and               α  r = 0 for all r ≥ 2. Proof: see [9]. Take  a n (x) = nk 0a nk  x k  and  bn(x) =  x n   Then the umbral composition c n (x) of polynomials sequences a n (x) = a nk x k  and  bn(x) = xn is written in the matrix form as : 0123( )( )( )( )( )nc xc xc xc xc x              = 0010 1120 21 2230 31 32 330 1 2 3 40 0 0 0 00 0 0 00 0 00 0n n n n n nnaa aa a aa a a aa a a a a a                0123nxxxxx           Now we discuss the effect of delta operator on umbral composition of polynomials set. The value from the Table 1 and operating Q on the above matrix, we get Q c 0 (x) = 0 Qc1 (x) = a11 α1 Qc2 (x)    =    α1 (a21 + 2 a22 x) + a22 α2                 =    α1  2121 1rrrra x      a2r xr-1 + a22 α2 Qc3(x)    =    α1 (a31 + 2 a32 x + 3 a33 x2) + α2 (a32 + 3 a33 x) + α3 a33 =α1 3131 1rrrra x     + α2  3232 2rrrra x      + α3 a33                   Qc n  (x)  =  α1 (an1 + 2 an2  x + 3 an3  x2 + …. + nC1 ann  xn-1)                                   + α2 (an2 + 3 an3  x + 12 an4  x2 + …. + nC2 ann  xn-2)                                   + α3 (an3 + 4 an4  x + 20 an5  x2 + …. + nC3 ann  xn-3)                                   + …. + α n  a nn. I S S N  2 3 4 7 - 1 9 2 1  V o l u m e  1 2  N u m b e r  0 7  J o u r n a l  o f  A d v a n c e s  i n  M a t h e m a t i c s  6397 | P a g e                                   c o u n c i l  f o r  I n n o v a t i v e  R e s e a r c h  J u l y  2 0 1 6                                                    w w w . c i r w o r l d . c o m                    Qc n  (x)  =    α1 11 1nrnrrra x     + α2 22 2nrnrrra x     + α3   33 3nrnrrra x                                                          + …. + α n  a nn.      Thus the effect of delta Q on the umbral composition of  nk 0a nk  x k  and   x n    can be   written in the matrix from as :  Q)()()()()(3210xcxcxcxcxcn=112221213 31 23 3 331 21 2 31 2 30 0 0 0 0 00 0 0 0 00 0 0 010 0 01 20 01 2 3rrrr rr rr rn n nr r rnr nr nr nnr r rara x ar ra x a x ar r ra x a x a x a                                                         n3210 Unbral Calculus is a powerful tool for dealing with recurrences. Recurrences play an important role in the theory of banks in signal processing. An important property of the Umbral Calculus is that it has its own generalization of Lagrange’s inversion formula. Thus we find many papers in which new forms of Lagrange’s inversion formula is derived using Umbral Calculus. 6.  STATUS AND FURTHER DIRECTIONS  The present attempt is to study the effect of the delta operator on umbral composition of sheffer sequence via sequential representation in equation (1) Theorem (2). In this study,  we consider the monomials {xn} for bn (x). The same theory may be extended to other  popular Sheffer polynomials such as falling factorial [x]n or raising factorial   [x] n instead of the monomials {xn}. References [1]    Di Bucchianico , A and  Loeb,  A Selected Survey of Umbral Calculus, Technische Universiteit  Eindhoven,                                              Department of Technology  Management, (2000). [2]     Di. Nardo, E ,  Niederhansen H and  Senato, D, A symbolic handling of Sheffer polynomials,  Annali Di Mathematica Pura Ed Applicata, Springer, September 2011. [3]    Email C.Popa. On an expansion theorem in the finite operator calculus of G.C.Rota, General Mathematics, Vol. 16, No. 4 (2008), 149-154. [4]    Ira M. Gessel, Applications of the Classical Umbral Calculus, Research article supported by NSF grand DMS-9972648, (2001)  [5]    JianhongShen, G.C. Rota and D. Tailor, All polynomials of binomial type are represented by Abel polynomials, MIT, Cambridge, USA, (1997). [6]    Maheswaran A and  Elango, C ,Sequential Representation of Delta Operator in Finite Operator Calculus, British Journal of Mathematics and Computer Science, 14(2): 1-11, 2016, DOI: 10.9734/BJMCS/2016/23323. [7]    Maheswaran A, and  Elango, C , On new identities for Basic Polynomials Sequences in Finite Operator Calculus, British Journal of Mathematics and Computer Science, 16(3):  2016, 1-11, DOI: 10.9734/BJMCS/2016/24896. [8]    Roman,  Steven  and  Rota G.C. , TheUmbral Calculus, Academic Press, New York      London, Feb 1978 [9]    Rota  G.C. ,  Finite Operator Calculus, Academic Press, London, 1975.  

Effect of Delta Operator on Umbral Composition in Finite Operator calculus

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Effect of Delta Operator on Umbral Composition in Finite Operator calculus

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