More on the Bernoulli and Taylor Formula for Extended Umbral Calculus

More on the Bernoulli*-T a ylor form ula for extended um bral calculus A.K.Kw a ´ sniewsk i High Sc ho ol of Mathematics and Applied Informatics PL - 15-021 Bialy stok , ul.Kamienna 17, P oland e-mail: kw andr@wp.pl August 17, 2021 Abstract One pre sen ts here the ∗ ψ -Bernoulli-T a ylor * formula of a new sor t with the r est ter m o f the Ca uc hy type recently der iv ed by the author in the ca se of the s o called ψ - difference calculus which co nstitutes the representative fo r the purp ose c ase of e xtended umbral ca lculus. The central impo rtance o f such a type formulas is b ey ond any doubt - and recent publications do confirm this histo rically establis hed exp eri- ence.Its links via umbralit y to combinatorics a re k no wn at leas t s ince Rota and Mullin-Rota source pa pers then up to r ecen tly extended by many authors to b e indicated in the sequel. KEY W ORDS: um bral calculus, Bernoulli formula,Gra v es-Heisen b erg- W eyl algebra(**) AMS S.C. (2000) 05A40, 81S05, 01A45, 01A50 , 01A61 * see b elo w : a h istorica l remark based on Academician N.Y.Sonin arti- cle published in Petersburg in 19-th cen tury . W e ow e this information and the article to Professor O.V.Visco v f rom Mosco w. ** s ee: C. Gra ves, On th e principles which r e gulate the inter change of symb ols in c e rta in symb olic e qu ations , Pr oc. Ro y al Ir ish Academ y 6 (1853– 1857) , 144-152 1 1 One Historical Rema rk Here are the famous examples of expan sion ∂ 0 = ∞ X n =1 x n − 1 n ! d n dx n or ǫ 0 = ∞ X n =0 ( − 1) n x n n ! d n dx n where ∂ 0 is the d ivided difference op erator while ǫ 0 is at the zero p oin t ev aluation functional. If one compares these with ”series univ ersalissim a” of J.Bernoulli from A cta Erudic orum (1694) (see commen taries in [12]) and with exp { y D } = ∞ X k =0 y k D k k ! , D = d dx , then confr on tation with B.T a ylor’s ”Metho dus incr ementorum dir e cta et in- versa” (1715 ), London; en titles o ne to call the e xpan sion formulas considered in this n ote ”Bernoul li - T aylor formulas” or (for n → ∞ ) ”Bernoul li - T ay- lor series” [1 ]. Information: Johann Bernoulli was elected a fello w of the aca demy of St Petersburg.Johann Bernoulli - the Disco verer of Series Universalis- sima was ” Arc himedes of his age ” and this is indeed inscrib ed on his tom bstone. 2 In tro duction A. F r om her e no w ψ denotes an extension of h 1 n ! i n ≥ 0 sequence to quite arbitrary one (the so called ”admissible”- see: Mark o wsky) and the sp ecific c hoices are for example : Fib onomialy -extended ( h F n i - Fib onacci sequence ) h 1 F n ! i n ≥ 0 or jus t ”the usual” h 1 n ! i n ≥ 0 or Gauss q -extended h 1 n q ! i n ≥ 0 admissible sequences of exte nd ed um b ral op erator cal culus - see more in [16, 13, 1 4, 15]. The simplicit y of th e first s teps to b e done while identifying general prop erties of su c h [6,7,8,1 6,13-15] ψ -extensions consists in natural n ota tion i.e. here - in writing ob jects of these extensions in mnemonic co nv enien t upside do wn notation [15] , [14] ψ ( n − 1) ψ n ≡ n ψ , n ψ ! = n ψ ( n − 1) ψ ! , n > 0 , x ψ ≡ ψ ( x − 1) ψ ( x ) , (1) 2 x k ψ = x ψ ( x − 1) ψ ( x − 2) ψ ... ( x − k + 1) ψ (2) x ψ ( x − 1) ψ ... ( x − k + 1) ψ = ψ ( x − 1) ψ ( x − 2) ...ψ ( x − k ) ψ ( x ) ψ ( x − 1) ...ψ ( x − k + 1) . (3) If one w rites the ab o ve in the f orm x ψ ≡ ψ ( x − 1) ψ ( x ) ≡ Φ( x ) ≡ Φ x ≡ x Φ , one sees that the n ame ups ide do wn notation is legitimate. Y ou may consu lt [1 0 ] , [9] ,[14,15] for further deve lopment and use of this notation. With su c h an extension w e frequent ly though not alw a ys ma y ” ψ -mnemonic” rep eat w ith exactl y the same simplicit y and b eaut y most of what was done b y Rot a [10, 15]. Accordingly the extension of notions a n d form ulas with its element ary essentia l con tent and conte xt to general case of ψ - um br al in- stead of um br al o r q -umbral ca lculi case only - is s omet imes automatic [10] , [14, 15] (see c orresp ond ing earlier references there and nece ssary definitions). B. While deriving the Bernoulli-T a ylor ψ -form ula one is tempted to adapt the ingenious Visko v‘s metho d [2] of arriving to form ulas of suc h t yp e for v arious pairs of op erations. In our case these w ould b e ψ -differentia tion and ψ -in tegration (see: App endix). Ho we ver straigh tforward ap plicat ion of Visco v metho ds in ψ -extensions of umbral calc ulu s leads to sequen ces whic h are not normal (W ard) hence a new inv en tion is needed. This exp ected and v erified here inv ention is the new sp ecific ∗ ψ pro duct of analytic functions or formal series. This note is based on [3] wher e the deriv ation of this new fo rm of Bernoulli-T a ylor ∗ ψ - formula w as d eliv ered due to the use of a sp ecific ∗ ψ pro duct of formal series. 3 Classical Bernoulli-T a ylor form ulas with the rest term of the Cauc h y t yp e b y V isk o v metho d Let us consider the ob vious identit y n X k =0 ( α k − α k +1 ) = α 0 − α n +1 (4) in whic h (4) w e no w p ut α k = a k b k ; a, b ∈ A . A is an asso ciativ e algebra with unit y ov er th e field F=R,C. Then we get n X k =0 a k (1 − ab ) b k = 1 − a n +1 b n +1 ; a, b ∈ A (5) 3 Numerous choic es of a, b ∈ A result in many imp ortan t sp ecifications of (5) Example 1. Let F denotes the linear s pace of sufficien tly smo oth fun c- tions f : F − → F . Let a : F − → F ; ( af ) ( x ) = Z b a f ( t ) d t, b : F − → F ; ( bf ) ( x ) = ( d dx f )( x ); (6) l : F − → F ; ( l f )( x ) = f ( x ) . Then [b,a]=1-ab= ε α where ε α is ev aluation functional on F i.e. ε α ( f ) = f ( α ) (7) Using no w the text-b o ok integ ral Cauc hy formula ( k > 0) ( a k f )( x ) = Z x a ( x − t ) k − 1 ( k − 1)! f ( t ) dt, (8) and un der th e choice (6) on e gets from (5 ) th e w ell-kno wn Bernoulli-T aylo r form ula f ( x ) = n X k =0 ( x − α ) k k ! f ( k ) ( α ) + R n +1 ( x ) (9) with the rest term R n +1 ( x ) in th e Cauch y form R n +1 ( x ) = Z x a ( x − t ) n n ! f ( n +1) ( t ) dt (10) Example 2. [1] Let F d enotes the linear space of functions f : Z + − → F ; Z + = N ∪ { 0 } . L et a : Z + − → F ; ( af )( x ) = x − 1 X k =0 f ( k ) , b : Z + − → F ; ( bf )( x ) = f ( x + 1) − f ( x ) , (11) l : Z + − → F ; ( lf )( x ) = f ( x ) . It is easy to see that [b,a]=1-ab= ε 0 where ε 0 is ev aluation fun ctio nal i.e. ε 0 ( f ) = f (0) . b = ∆ is the stand ard difference op erator with its left in verse 4 definite sum matio n op erator a. The corresp ondin g ∆ - calculus Cauc hy form ula is also kn o wn (see formula (31 p.310 in [5 ]); ( a k f )( x ) = x − 1 X r =0 ( x − r − 1) k − 1 ( k − 1)! f ( r ); k > 0 (12) where x n = x ( x − 1)( x − 2) ... ( x − n + 1) . Under the c hoice (11 ) one gets from (5) the ∆ - calculus Bern oulli - T aylo r fom ula [1] f ( x ) = n X k =0 x k k ! (∆ k f )(0) + R n +1 ( x ) (13) with the rest term R n +1 ( x ) in th e Cauch y ∆ form R n +1 ( x ) = x − 1 X r =0 ( x − r − 1) n n ! (∆ n +1 f )( r ); (14) 4 ” ∗ ψ realization” of Bernoulli iden tit y . No w a sp e cific al ly new form of th e Bernoulli-T aylo r formula with the rest term of th e Cauc hy t yp e as w ell as Bernoulli-T a ylor series is to b e su pplied in t h e case of ψ -difference umbr al calculus (see [5-8] a nd [9,10 ] and references therein). F or that to do we use natural ψ -umbral r epresen tation [13,14] of Gra v es-Heisen b erg-W eyl (GHW) algebra [1 1,12] generators ˆ p and ˆ q and then w e use Bernoulli identit y (15) ˆ p n X k =0 ( − ˆ q ) k ˆ p k k ! = ( − ˆ q ) n ˆ p n +1 n ! (15) deriv ed by Visko v from (4 ) under the substitution (see (28) in [2]) α 0 = 0 , α k = ( − 1) k ( ˆ q ) k − 1 ˆ p k ( k − 1)! , k = 1 , 2 , ... due to ˆ p ˆ q n = ˆ q n ˆ p + n ˆ q n − 1 (n=1,2,...) resulting by ind uction from [ ˆ p, ˆ q ] = 1 (16) Example 1. T he choice ˆ p = D ≡ d dx and ˆ q = ˆ x − y , y ∈ F ; ˆ xf ( x ) = xf ( x ) after substitution in to Bernoulli identit y (15) and in tegration R x α dt giv es the Bernoulli - T a ylor formula (9 ). 5 Example 2. The c hoice [2] ˆ p = ∆ and ˆ q = ˆ x ◦ E − 1 where E α f ( x ) = f ( x + α ) after substitution into Bernoulli identi ty (15) and ”∆ - integrat ion” P α − 1 r =0 giv es the Be rn oulli - Ma c laurin form u la of t h e follo wing form ( α, x ∈ Z , ▽ = 1 − E − 1 ) with the rest term R n +1 ( x ) f (0) = n X k =0 α k k ! ( − 1) k +1 ( ▽ k f )( α ) + R n +1 ( α ); (17) R n +1 ( α ) = ( − 1) n α − 1 X r =0 r n n ! ( ▽ n +1 f )( r + 1) . (18) Example 3. Here f ( k ) ≡ ∂ k ψ f and f ( x ) ∗ ψ g ( x ) ≡ f ( ˆ x ψ ) g ( x ) - see App endix. The choice ˆ p = ∂ ψ and ˆ q = ˆ z ψ ( z = x − y ) w here ˆ x ψ x n = n +1 ( n +1) ψ x n +1 after substitution in to Bernoulli iden tit y (15) and ” ∂ ψ - integ ration” R x α d ψ t (see: App endix) give s another Bernoulli - T a ylor ψ -form ula of the form: f ( x ) = n X k =0 1 k ! ( x − α ) k ∗ ψ ∗ ψ f ( k ) ( α ) + R n +1 ( x ) (19) with the rest term R n +1 ( x ) in th e Cauch y-form R n +1 ( x ) = 1 n ! Z x α d q t ( x − t ) n ∗ q ∗ q f ( n +1) ( t ) dt (20) In the abov e notation x 0 ∗ ψ = 1 , x n ∗ ψ ≡ x ∗ ψ ( x ( n − 1) ∗ ψ ) = x ∗ ψ ... ∗ ψ x = n ! n ψ ! x n ; n ≥ 0 . Naturally ∂ ψ x n ∗ ψ = nx ( n − 1) ∗ ψ and in general f , g - ma y b e f ormal s eries for whic h ∂ ψ ( f ∗ ψ g ) = ( D f ) ∗ ψ g + f ∗ ψ ( ∂ ψ g ) (21) i.e. Leibniz ∗ ψ rule holds [13, 14, 15]. Summary: These another forms of b oth the Bernoulli -T a ylor f orm u la with the rest term of the Cauc hy type [3 ] as well as Bern oulli - T a ylor s eries are quite easily handy due to the technique dev elop ed in [13 , 14] where one may find more on ∗ ψ pro duct devised p erfectly suitable for the W ard ’s ”c alculus of se quenc es” [6] or more exactly ∗ ψ is d evised p erfectly suitable f or the so- called ψ - extension on Finite Op erator Calculus of Rota (see [9, 10, 14, 15] and references therein) 6 5 App endix ∗ ψ pro duct Let n − ψ ≡ ψ n ; ψ n 6 = 0: n > 0. Let ∂ ψ b e a linear operator acting on formal series and defined accordingly b y ∂ ψ x n = n ψ x n − 1 . W e introd uce no w a in tuition app ealing ∂ ψ -difference-izati on ru les for a sp e- cific new ∗ ψ pro duct of functions or formal series. This ∗ ψ pro duct is what w e call: the ψ -m ultiplication of fu nctions or formal series as s p eci fi ed b elo w . Notation A.1. x ∗ ψ x n = ˆ x ψ ( x n ) = ( n +1) ( n +1) ψ x n +1 ; n ≥ 0 hence x ∗ ψ 1 = (1 ψ ) − 1 x 6≡ x therefore x ∗ ψ α 1 = α 1 ∗ ψ x = x ∗ ψ α = α ∗ ψ x = α (1 ψ ) − 1 x and ∀ x, α ∈ F ; f ( x ) ∗ ψ x n = f ( ˆ x ψ ) x n . F or k 6 = n x n ∗ ψ x k 6 = x k ∗ ψ x n as well as x n ∗ ψ x k 6 = x n + k - in general. In ord er to facilitate the form ulation of observ ations accounte d for on the basis o f ψ -calculus represen tation of GHW algebra w e shall use what follo w s. Definition A.1. With Notation A.1. ad opted defin e the ∗ ψ p o w ers of x according to x n ∗ ψ ≡ x ∗ ψ x ( n − 1) ∗ ψ = ˆ x ψ ( x ( n − 1) ∗ ψ ) = x ∗ ψ x ∗ ψ ... ∗ ψ x = n ! n ψ ! x n ; n ≥ 0. Not e th at x n ∗ ψ ∗ ψ x k ∗ ψ = n ! n ψ ! x ( n + k ) ∗ ψ 6 = x k ∗ ψ ∗ ψ x n ∗ ψ = k ! k ψ ! x ( n + k ) ∗ ψ for k 6 = n and x 0 ∗ ψ = 1. This noncomm utativ e ψ -pro duct ∗ ψ is d evised so as to ensure the fol- lo wing obser v ations. Observ ation A.1 a) ∂ ψ x n ∗ ψ = nx ( n − 1) ∗ ψ ; n ≥ 0 b) exp ψ [ α x ] ≡ exp { α ˆ x ψ } 1 c) exp [ αx ] ∗ ψ ( exp ψ { β ˆ x ψ } 1 ) = ( exp ψ { [ α + β ] ˆ x ψ } ) 1 d) ∂ ψ ( x k ∗ ψ x n ∗ ψ ) = ( D x k ) ∗ ψ x n ∗ ψ + x k ∗ ψ ( ∂ ψ x n ∗ ψ ) e) ∂ ψ ( f ∗ ψ g ) = ( D f ) ∗ ψ g + f ∗ ψ ( ∂ ψ g ) ; f , g - formal series f ) f ( ˆ x ψ ) g ( ˆ x ψ ) 1 = f ( x ) ∗ ψ ˜ g ( x ) ; ˜ g ( x ) = g ( ˆ x ψ ) 1 . Um bral ” ∼ ” Note: ˜ g ( x ) = g ( ˆ x ψ ) 1 defines the map ∼ : g 7→ ˜ g i.e. ∼ : P 7→ P whic h is an umbral op erator. ψ -In tegration Let: ∂ o x n = x n − 1 . The linear op erator ∂ o is id en tical with divided difference op erator. Let ˆ Qf ( x ) f ( q x ). Re call a lso th at to the 7 ” ∂ q difference-izatio n” there corresp ond s the q -in tegration which is a right in ve rs e op eration to ” q -difference-ization”. Namely F ( z ) : ≡  Z q ϕ  ( z ) := (1 − q ) z ∞ X k =0 ϕ  q k z  q k (22) i.e. F ( z ) ≡  Z q ϕ  ( z ) = (1 − q ) z ∞ X k =0 q k ˆ Q k ϕ ! ( z ) = =  (1 − q ) z 1 1 − q ˆ Q ϕ  ( z ) . (23) Of course ∂ q ◦ Z q = id (24) as 1 − q ˆ Q (1 − q ) ∂ 0  (1 − q ) ˆ z 1 1 − q ˆ Q  = id. (25) Naturally (25) migh t serv e to define a right in v erse op eration to ” q -difference- ization” ( ∂ q ϕ ) ( x ) = 1 − q ˆ Q (1 − q ) ∂ 0 ϕ ( x ) and consequen tly the ” q -in tegratio n ” as represent ed by (22) and (23 ). As it is wel l kno wn th e d efinite q -in tegral is an numerical appr o ximation of the defin ite integ ral obtained in the q → 1 limit. Finally we in tro duce the analogous representa tion for ∂ ψ difference-izatio n ∂ ψ = ˆ n ψ ∂ o ; ˆ n ψ x n − 1 = n ψ x n − 1 ; n ≥ 1 (26) Then Z ψ x n =  ˆ x 1 ˆ n ψ  x n = 1 ( n + 1) ψ x n +1 ; n ≥ 0 (27) and of course  R ψ ≡ R d ψ  ∂ ψ ◦ Z ψ = id (28) Naturally ∂ ψ ◦ Z x a f ( t ) d ψ t = f ( x ) 8 The formula of ”p er p artes” ψ -in tegration is easily obtainable from (Obser- v ation A.1 e) and it reads: Z b a ( f ∗ ψ ∂ ψ g )( t ) d ψ t = [( f ∗ ψ g )( t )] b a − Z b a ( D f ∗ ψ g )( t ) d ψ t (29) Closing Remarks: I. All th ese ab o v e may b e quite easily extended [15] to the case of an y Q ∈ E nd ( P ) linear op erator that reduces by one the degree of eac h p olynomial [16]. Namely one introdu ces [15]: Definition A.2. ˆ x Q ∈ E nd ( P ) , ˆ x Q : F [ x ] → F [ x ] suc h th at ( x n ) = ( n +1) ( n +1) ψ q n +1 ; n ≥ 0; where Qq n = n q n − 1 . Th en ⋆ Q pro duct of formal series and Q -in tegratio n are defined almost mnemonic analo gously . I I. In 1937 Jean Delsarte [17] had deriv ed the general Bernoulli-T aylor form ula fo r a class of linear op erators δ including linear op erators that redu ce b y one the degree of eac h p olynomial. The rest term of the Cauch y-like t yp e in his T a ylor formula (I) is give n in terms of the uniqu e solution of a first order partial d ifferen tial equation in t w o real v ariables. This first order partial differen tial equation is determined by the c hoice of the linear op erator δ and the function f under expansion. In our Bernoulli-T ayl or -form ula (16)- (17) or in its str aig htforw ard ⋆ Q pro duct of form al series and Q -in tegration generalizat ion - there is no need to solv e any p artia l d ifferen tial equation. I I I. In [18] (1941) Pr ofessor J. F. Steffensen - th e Master of p olynomials application to actuarial problems ( see : h ttp://www.math.ku.dk/arkivet /jfsteff/stfarkiv.ht m ) supplied a r emarcable deriv ation of another Bernoulli-T a ylor form ula with the rest of ”Q-Cauc h y t yp e” in the example presenting the ”Ab el p o w eroids” IV. The recen t p ap er [19] (2003) by Mourad E. H. Ismail, Denis Stan- ton ma y s erv e as a kind of indication for p ursuing further in ve stigation. There the authors ha ve established tw o new q-analogues of a T a ylor se- ries expansion for p olynomia ls using sp ecial Ask ey-Wilson p olynomial bases. As ”bypro ducts” their imp ortan t pap er includes also new summ at ion theo- rems,quadratic tr ansformations for q-series and new results on a q-exp onen tial function. V. Let us also dra w an atten tion to t wo more differen t publication on the sub ject wh ic h are the ones referr ed to as [20,21].The q -Bernoulli theo rems 9 are n amed here and there ab o ve as q -T a ylor theorems. The corresp onding ( q , h )-Bernoulli theorem for the ∂ q ,h -difference calculus of Hahn [22] m igh t b e also obtained as the the one ( q , 0)-Bernoulli i.e. q -Bernoulli theorem constituting here the sp ecial case the Visk o v method [2] application. Th is is so b ecause the ∂ q ,h -difference calculus of Hahn [22] may be redu ced to q -calculus of Thomae-Jac kson [5,23] due to the follo wing observ ation . Let h ∈ R, ( E q ,h ϕ )( x ) = ϕ ( q x + h ) and let ( ∂ q ,h ϕ )( x ) = ϕ ( x ) − ϕ ( q x + h ) (1 − q ) x − h (30) Then (see Hann [5]and [22]) ∂ q ,h = E 1 , − h 1 − q ∂ q E 1 , h 1 − q . (31) Due to (30) it is easy no w to deriv e corresp onding form u las including Bernoulli- T a ylor ∂ q ,h -form ula obtained in [24] by the Visko v metho d [2] which for q → 1 , h → 0 reco v ers the con ten t of one of th e examples in [2] , w hile for q → 1 , h → 1 one r eco v ers the con tent of th e another example in [2]. Th e case h → 0 is included in the form ulas o f q -calculus o f Thomae-Jac kson easy to be sp ecified : see [22] (see also thousands of up-date references there). F or Bernoulli- T a ylor F orm ula (presen ted during P T M - Conv ention Lo dz - 2002) : con tact [23] for its recen t v ersion. The comparison of the all abov e qu ote d w a ys to arriv e at extended Bernoulli form ulae we lea v e for another exhib itio n of similar inv estigations. VI. As indicated righ t after O bserv ation A.1.e) the rule ˜ g ( x ) = g ( ˆ x ψ ) 1 defines the map ∼ : g 7→ ˜ g which is an umbral op erator ∼ : P 7→ P . It is mnemonic extension of the corresp onding q - definition b y Kirsc henh ofer [24] and Cigler [25]. This umbral op erator (without reference to to [24,25]) had b een already us ed in theoretical ph ysics aiming at Quant um Mec hanics on the lattic e [26]. Th e simila r aim is r epresen ted by [27,28] (see fu rther ref- erences there) where in cidence algebras are b eing p repared for that purp ose (Dirac notation included). As it is well kno wn - the classical um br al [29,30] and extend ed [10] finite op erator calculi m a y b e form ulated in th e reduced 10 incidence algebra language. Hence b oth applications of related to ols to the same goal are exp ected to meet at the arena of GHW algebra description of b oth [14,13]. References [1] N. Y. S onin Riad Ivana Be rnoul li (in ol d russion) Bul letin de l‘A c ad ´ emie Imp ´ erial des Scienc es de St.-P´ etersb our g 7 No.4(1897), 337-353 [2] 0.V. Visk o v: Nonc ommutative Appr o ach to Classic al Pr oblems of Anal- ysis (in russ ion) T rudy M atiematicz‘eskovo Instituta AN SSSR 177 , 21 (1986 ). [3] A. K. Kwa ´ sniewski: q-differ enc e c alculus Bernoul li- T aylor formula Bia lystok Univ. In st. Comp. Sci. UwB Pr eprin t 32 (Augu st 2001) [4] A. O. 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