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Umbral Methods, Combinatorial Identities and Harmonic Numbers Applied Mathematics. 2011; 1(1): 46-49 DOI: 10.5923/j.am.20110101.06 Umbral Methods, Combinatorial Identities and Harmonic Numbers K. V. Zhukovsky1,*, G. Dattoli2 1Faculty of Physics, Moscow State University, Leninskie Gory, Moscow 119899, Russia 2ENEA Research Centre, Fis-Mat, 00044 Frascati, Rome, Italy Abstract We analyse and demonstrate how umbral methods can be applied for the study of the problems, involving combinatorial calculus and harmonic numbers. We demonstrate their efficiency and we find the general procedure to frame new and existent identities within a unified framework, amenable of further generalizations. Keywords Umbral, Identities, Calculus, Harmonic Numbers 1 aˆs aˆ r11 aˆ s r (6) 1. Introduction sr1 b) The ―addition‖ theorem: In this article we employ methods of umbral nature to mn 11psmn provide a common framework for known and new identities ( 1) ( 1) (7) m n 11ps p s regarding combinatorial calculus and harmonic numbers. ps00 Just to give a glimpse into the technique, adopted in this c) The multiplication theorem: 1 n n article, we remind the identity [1] sm ( 1) ... n mn1 s 0 sm 11s n m ( 1) (1) (8) n n ns11s0 s s1 1 ( 1) ,mm intrger 0 s m which, after defining the umbral variable aˆ (see[2]), s1 0 1 1 sr where 1 — the vacuum state of the space, on which the r1 operator aˆ acts, and 1ˆ — the unit operator: The proof of b) and c) theorems is achieved by the same s 1 0 procedure leading to the proof of a) and is omitted here for aaˆ 1 ,ˆ 1ˆ (2) 1 s 1 the sake of conciseness can be cast in the following form: Now let us introduce the operator of the umbral deriva- tive, defined by the following rule: 1 n (1aˆ ) 1. (3) ˆ nn1 n 1 a aˆ naˆ (9) Equation (1) can be written in the form (3) just as the which, along with the multiplication condition: consequence of the binomial theorem and of definition (2) aˆ aˆnn aˆ 1 (10) and it is a useful tool to generate new identities, listed be- yields the following result for the commutator bracket low: between the two operators: a) The duplication ―theorem‖: ˆ ,1aˆ ˆ aˆ aˆ ˆ ˆ , (11) nn a a a 11rsn ( 1) ( 1) . (4) Equation (11) ensures that, we can benefit from the prop- 2n 1rs00r s r 1 erties of the Weyl-Heisenberg algebra, characterising our The proof of this last identity is easily achieved by fol- problem. Within this framework the following simple ex- lowing the steps outlined below. We can obtain the obvious ample is provided by the definition of the associated Her- consequence of the equation (3): mite polynomials: two variables Hermite polynomials are nn 1 2n n n rnn r s s (1 aˆ )1(1 aˆ )(1 aˆ )1 (1) aˆ (1) aˆ 1(5) defined below with the variable x replacing the operator aˆ 21n rs00rs as follows2: n1 thus getting equation (4) from the identity 1 H( aˆ , y )1 H (1, y ) y2 nnn 1 1 * Corresponding author: , nn [email protected] (K. V. Zhukovsky) 22 aˆn2 r y r y r Published online at http://journal.sapub.org/am (12) Hn ( aˆ , y )1 n ! 1 n ! Copyright © 2011 Scientific & Academic Publishing. All Rights Reserved rr00(n 2)!! r r ( n 2)!( r n 2 r 1)! r Applied Mathematics. 2011; 1(1): 46-49 47 It is easy to show that the following recurrences are satis- Therefore, we can reformulate all the theorems a) – c) in fied: a fairly direct way in terms of Beta function. For example, ˆ (aH n ( aˆ , y ))1 nH n1 ( aˆ , y )1 (13) the duplication identity can be written as follows: mm and 1 rs mm ˆ 2 ( 1) B ( 1, r s 1) (23) (yH n ( aˆ , y ))1 a H n ( aˆ , y )1 (14) 21m rs00 rs which are direct generalisations of the relevant to the or- Taking repeated derivatives of both sides of (19) with dinary Hermite polynomials relations. The umbral heat respect to λ in the point λ = 0 yields the r power of the equation (14) can be exploited to define the polynomials left-hand side of equation (18), written in terms of the Beta (12) in terms of the following operational equation: function instead of the Stirling numbers: ˆ ˆ 2 n 1m m Bs(r 1) (1, 1) Hna( a , y )1 exp y a 1 (15) sr r ( 1) (24) In these introductory remarks we have presented few (m 1) s0 s (r 1)! elements of the formalism, which we employ in the follow- where ()rr ing chapters to further develop the method of umbral op- B(1, s 1) B ( 1, s 1) | . (25) erators and obtain new identities in combinatorial calculus, 0 involving the Euler Beta and Riemann Zeta functions. This last result (24) represents essentially the equation (18), written without any explicit use of infinite sums. As to the explicit evaluation of the derivatives of the Beta func- tion, we note that they possess the integral representation, 2. Umbral Methods and the Euler Beta provided by[6],[7]: Function 1 B( x , y ) (1 t )xy11 t dt (26) Let us take note that the parameter n in the equation (1) 0 can be treated as a variable and, therefore, p times repeated and, therefore, we find the following expression for the derivatives with respect to n can be taken on both sides: derivative of the Beta function: m 1 1 ( 1) m ()rsr ˆ ˆ n B(1, s 1) ln(1 t ) t dt (27) p1 ln(1 aa ) (1 ) 1 (16) (n 1) m! 0 Now, using the following series expansion [4]: In the next chapter we will apply the above obtained re- S(,) k m sults to the theory of the harmonic numbers. [ln(1x )]mk m ! ( x ) (17) km k! assumed to be valid also for the umbral operator , we end up with the following identity: 3. Umbral Methods and Harmonic m n 1 ( 1)S ( k , m )ks n 1 Numbers m1 ( 1) (18) (n 1) m!k m k ! s 0 s s m 1 The harmonic numbers[8] are usually denoted by Hn; to where S k, m — the Stirling numbers of the first kind avoid confusions with Hermite polynomials we use here hn [4]. The validity of (18) has been checked aposteriori by a notation: numerical procedure. Explicit study of the Stirling numbers n1 1 relations with combinatorial identities can be found in[5]. hhn ,00 (28) r 1 Note, that identity (18) involves infinite sums and they r0 can be avoided, if we rewrite (18) with the help of the iden- Umbral methods technique simplifies the derivation of tity the properties of the harmonic numbers and the study of the m associated generating functions. 1 s m ( 1)Bs ( 1, 1) (19) We can express the harmonic numbers (28) in terms of m 1 s0 s the umbral variable (2) as follows: where B(,) x y — the Euler Beta function, which writes nn111 (1aˆ )n n ˆ r rˆ r in terms of the Euler Gamma function as follows[6]: hn (1 a ) 1 1 ( 1) a 1 (29) rr01aˆ r ()()xy B(,) x y . (20) ()xy and derive the following relation between the harmonic numbers and the umbral variable: We proceed on the assumption that the above definition 1 (1 aaˆ )nn 1 (1 ˆ ) (20) is valid also for the umbral variable to write (19) in ˆ n aˆ ha2n 1 1 (1 ) 1 (30) the following form: aaˆ ˆ 1 which eventually yields: ˆ n (1 a ) 1 (21) nn nn m 1 sr1 1 hh2nn ( 1) (31) where we denoted the power of the umbral variable rr01 sr sr via the beta function as follows: Further extensions can be easily obtained without con- s aˆ 1 B ( 1, s 1) (22) ceptual difficulties, except for some cumbersome algebraic 48 K. V. Zhukovsky et al.: Umbral Methods, Combinatorial Identities and Harmonic Numbers steps, for example: n n A( n , y ) h yjn (1 h y ) (44) nn 2nn 1 j rs1 j0 j ( 1) hh32nn (32) rr01 sr sr Then from the definition of the higher moment we ob- Analogous relations can be obtained for the generalized tain: harmonic numbers, defined as follows: n ()m mn m n1 A( n ) hj ( y y ) A0, n ( y ) | y 1 ()()mm1 hh,0 (33) r0 j n m 0 (45) r0 (r 1) m kk The use of the above described procedure and of the S2( m , k ) yy A 0, n ( y ) | y 1 identities derived in the previous chapter 2 yields the gener- k 1 alization of the formula (32): where S2(,) m k — the Stirling numbers of the second nn( 1)r sm 1 2nn kind. The A(,) n y can be obtained via the same procedure, B(m 1)(1, s r 1) h ( m ) h ( m ) (34) 32nn which yielded (40), namely: rr01(m 1)! sr n m ( 1) n n m m Further comments on umbral methods and harmonic A( n , y ) (1 y ) y (46) numbers are given in the following concluding chapter. m1 m m Eventually we obtain the high order moments n m (1) ( 1) nm n nm 4. Generalisations and Discussion An( ) 2 (47) m1 m m 2 To complete the study of the harmonic numbers and um- and bral methods, consider the following sum: n m 2 (2) ( 1)nm n (n m ) n m n An( ) 2 n m m 4 m1 (48) A() n hj (35) j0 j 1 (1 n ) A ( n ) (2n 1)(2 n 1) n Employing the results of the previous chapter 3 we recast 4 A(n) in the following operator form: Higher than 2d order moments can be derived using the A( n ) (1 h )n 1 (36) Stirling numbers and other new identities can be generated with the help of the Matematica software.
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