New identities for the partial Bell polynomials Djurdje Cvijovi´c Atomic Physics Laboratory, Vinˇca Institute of Nuclear Sciences P.O. Box 522, 11001 Belgrade, Republic of Serbia 3 E-Mail: [email protected] 1 Abstract. A new explicit closed-form formula for the multivariate 0 2 (n,k)th partial Bell polynomial Bn,k(x1,x2,...,xn−k+1) is deduced. n The formula involves multiple summations and makes it possible, for a thefirsttime,toeasilyevaluateB directlyforgivenvaluesofnand n,k J k (n ≥ k,k = 2,3,...). Also, anew addition formula (with respectto 6 k) is found for the polynomials B and it is shown that they admit 1 n,k a newrecurrencerelation. Several specialcases andconsequences are ] pointed out, and some examples are also given. A C 2010 Mathematics Subject Classification. Primary 11B83, 11B75, 11B99; . Secondary 11B73, 11B37. h t a Key Words and Phrases. Partial Bell polynomial; Recurrence relation; Stirling number m of thesecond kind. [ 1. Introduction 1 v For n and k non-negative integers, the (exponential) (n,k)th partial Bell polyno- 8 5 mial in the variables x1,x2,...,xn−k+1 denoted by Bn,k ≡ Bn,k(x1,x2,...,xn−k+1) 6 may be defined by the formal power series expansion (see, for instance, [1, pp. 133, 3 Eq. (3a’)]) . 1 ∞ k ∞ 0 1 tm tn 3 xm = Bn,k(x1,x2,...,xn−k+1) (k ≥0), (1.1) k! m! n! 1 m=1 ! n=k X X : v or, what amounts to the same, by the explicit formula [2, p. 96] i X n! x1 ℓ1 x2 ℓ2 xn−k+1 ℓn−k+1 B = ... , (1.2) r n,k a ℓ1!ℓ2!...ℓn−k+1! 1! 2! (n−k+1)! (cid:18) (cid:19) X (cid:16) (cid:17) (cid:16) (cid:17) where (multiple) summation is extended over all partitions of a positive integer number n into exactly k parts (summands), i.e., over all solutions in non-negative integers ℓ , 1 ≤ α ≤n−k+1, of a system of the two simultaneous equations α ℓ1+2ℓ2+···+(n−k+1)ℓn−k+1 = n and ℓ1+ℓ2+···+ℓn−k+1 = k. For fixed n and k, B has positive integral coefficients and is a homogenous and n,k isobaric polynomial in its (n−k+1) variables x1,x2,...,xn−k+1 of total degree k and total weight n, i.e., it is a linear combination of monomials xℓ1xℓ2...xℓn−k+1 1 2 n−k+1 whose partial degrees and weights are constantly given by ℓ1+ℓ2+...+ℓn−k+1 = k and ℓ1+2ℓ2+...+(n−k+1)ℓn−k+1 =n. For some examples of these polynomials see Section 3. 2 Djurdje Cvijovi´c The partial Bell polynomials are quite general polynomials, they have a number of applications and more details about them can be found in Bell [3], Comtet [1, pp. 133–137], Hazewinkel[2,pp. 95–98], Charalambides[4,pp. 412–417]andAldrovandi [5, pp. 151–182]. However, the following formulae for B n,k n−k 1 1 n n+1 Bn,k = · (k+1)− xα+1Bn−α,k, (1.3) x1 n−k α α+1 α=1(cid:18) (cid:19)(cid:20) (cid:21) X n k1!k2! n Bn,k1+k2 = (k1+k2)! α Bα,k1Bn−α,k2 (1.4) α=0(cid:18) (cid:19) X and k 1 n−1 α1−1 αk−1−1 n α1 αk−1 Bn,k+1 = ··· ··· (k+1)! αX1=k α2X=k−1 αXk=1 z(cid:18)α1(cid:19)(cid:18)α2(cid:19)}| (cid:18) αk (cid:19){ k ·xn−α1x|α1−α2···x{αzk−1−αkxαk} (n ≥ k+1,k = 1,2,...) (1.5) appear not to have been noticed in any work on the subject which we have seen. In this note it is aimed to provide short proofs of these results, show some immediate consequences of them and provide some application examples (see also Section 3). 2. Proof of the main results We begin by showing that the identity (1.3) follows without difficulty from the definition of partial Bell polynomials B by means of the generating relation (1.1), n,k given that the next auxiliary result for powers of series is used. Consider ∞ k ∞ f xn = g (k)xn. (2.1) n n ! n=1 n=k X X For a fixed positive integer k, we have that: g (k) = fk, (2.2a) k 1 n−k 1 gn(k) = (α+1)(k+1)−(n+1) fα+1gn−α(k) (2.2b) (n−k)f1 αX=1h i (n ≥ k+1). Indeed, by comparing (2.1) with the definition of B in (1.1) and upon setting n,k g (k) = k!B /n! and f = x /n!, we arrive at the proposed formula (1.3) by n n,k n n utilizing (2.2b). Note that (2.2b) may be found in the literature (see [6]) but it is not as widely known(andevenlessused)asitshouldbe. Itisexactlyforthisreasonthatwederive itstartingfromthefollowingmoregeneral(andequallylittleknown)recurrencerela- tioninvolvingtheseriescoefficientsf andg (k)in( ∞ f xn)k = ∞ g (k)xn. n n n=0 n n=0 n n P P α(k+1)−n fαgn−α(k) = 0 (n ≥ 0). (2.3) α=0 X(cid:2) (cid:3) New identities for the partial Bell polynomials 3 k First, upon taking logarithms of each side of the equation g(x) = f(x) and ′ then differentiating both sides of the result with respect to x, we obtain f(x)g (x) = ′ (cid:2) (cid:3) kf (x)g(x). Next, insert the power series expansions of the various functions in this equation and multiply both sides by x, to get ∞ ∞ ∞ ∞ f xm· mg (k)xm = k mf xm· g (k)xm. (2.4) m m m m m=0 m=0 m=0 m=0 X X X X ∞ ∞ Now,recallthatif a and b aretwoseries,thentheirCauchyproduct m=0 m m=0 m ∞ n is the series n=0cn where cn = k=0akbn−k. This is to say that in the particular caseat hand,by eqPuatingthecoePfficients of agiven power of x, say xn,onbothsides of (2.4), we Phave nα=0(n−α)fαPgn−α(k) = k nα=0αfαgn−α(k), which eventually gives (2.3). The recurrence relation (2.3) is clearly valid for an arbitrary real or P P complex number k and it can be used to compute successively as many of the unknown gm(k) values as desired, in order g0(k),g1(k),g2(k),..., if g0(k) is known. Thespecialcase of(2.3) solved forgn(k), fork apositive integer andf0 6= 0,appears in various editions of the standard reference book by Gradshteyn and Ryzhik (see, for instance, [7, p. 17, Entry 0.314]) Finally,ifwesupposef0 = 0andf1 6= 0then,from( ∞n=0fnxn)k = ∞n=0gn(k)xn, wherek isapositiveinteger, itisobviousthatthecoefficientg (k),n = 0,1,...,k,is n only nonzero when n = k, g (k) then equals fk (see (P2.2a)), while ( P∞ f xn)k = k 1 n=0 n ∞n=0gn(k)xn reduces to (2.1). Therefore, since g0(k) = g1(k) = ... = gk−1(k) = 0 P and f0 = 0, the recurrence relation (2.3) becomes P n−k α(k+1)−n fαgn−α(k) = 0 (n ≥ k), α=1 X(cid:2) (cid:3) so that, upon replacing n by n+1, putting α+1 for α and solving for g (k), we n have that the coefficients g (k), n≥ k+1, are given by (2.2b) above. n In order to prove (1.4) we shall again resort to the generating relation for B n,k (1.1). Letusby[tn]φ(t)denotethecoefficientoftn inthepowerseriesofanarbitrary φ(t). Put f(t)= ∞ x tm, then by (1.1), we have m=1 m m! P k1!Bn,k1 = n![tn]f(t)k1 (n ≥ k1) and k2!Bn,k2 = n![tn]f(t)k2 (n ≥ k2), thus (k1 +k2)!Bn,k1+k2 = n![tn]f(t)k1+k2 = n![tn] f(t)k1 ·f(t)k2 n (cid:16)n (cid:17) = n! [tα]f(t)k1 ·[tn−α]f(t)k2 = n! k1!Bα,k1 k2!Bn−α,k2, (2.5) α! (n−α)! α=0 α=0 X X (n ≥ k1+k2) since [tn] φ(t)ψ(t) = n [tα]φ(t) · [tn−α]ψ(t) (the Cauchy product of two se- α=0 ries). We(cid:16)conclude(cid:17)the proof by noting that the required expression (1.4) follows by P rewriting (2.5). 4 Djurdje Cvijovi´c Lastly, we shall prove the closed-form formula (1.5) by making use of (1.4). It suffices to show that the addition formula for B (1.4) may be used to deduce the n,k following: n−1 1 n Bn,2 = xn−αxα (n ≥ 2), (2.6) 2! α α=1(cid:18) (cid:19) X n−1 α−1 1 n α Bn,3 = xn−αxα−βxβ (n ≥ 3) (2.7) 3! α β α=2β=1(cid:18) (cid:19)(cid:18) (cid:19) X X and n−1 α−1β−1 1 n α β Bn,4 = xn−αxα−βxβ−γxγ (n ≥ 4). (2.8) 4! α β γ α=3β=2γ=1(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) X X X By bearing in mind that Bn,1 = xn (this is a simple consequence of the definition Bn,k in (1.1)) and upon noticing that x0 = 0 (again, see (1.1)), the expression for Bn,2 given in (2.6) follows by (1.4) with k1 = 1 and k2 = 1. Further, this result for Bn,2 together with (1.4), where k1 = 2 and k2 = 1, leads to (2.7). It is clear that by repeating this procedure recursively we may obtain Bn,4, and so on. 3. Further results and concluding remarks Weremarkthattheexplicitclosed-formformulaforBn,k(x1,x2,...,xn−k+1)given by(1.5)isparticularlyuseful. Namely,itishardtoworkwiththeformula(1.2)which explicitly defines B due to complicated multiple summations, and, for instance, n,k it is virtually impossible by its use to write down a polynomial for given values of n and k. However, formula (1.5), although also involves multiple summations, makes this possible. In other words, it is now possible to directly evaluate B for given n n,k and k (n ≥ k,k = 2,3,...) by utilizing (1.5) instead of computing it recursively by making use of some recurrence relations (see, for instance, (1.3)). It is noteworthy to mention that the practical evaluation is greatly facilitated by wide availability of various symbolic algebra programs. In order to demonstrate an application of this result, we list several of the polynomials B determined by the formula (1.5), n,k where all the computations were carried out by using Mathematica 6.0 (Wolfram Research) 6 5 2 6 B8,7 =28x1x2, B9,7 =378x1x2+84x1x3, 4 3 5 6 B10,7 =3150x1x2+2520x1x2x3+210x1x4, 3 4 4 2 5 2 5 6 B11,7 =17325x1x2+34650x1x2x3+4620x1x3+6930x1x2x4+462x1x5, 2 5 3 3 4 2 4 2 B12,7 =62370x1x2+277200x1x2x3+138600x1x2x3+103950x1x2x4 5 5 6 +27720x1x3x4+16632x1x2x5+924x1x6, 6 2 4 3 2 2 4 3 B13,7 =135135x1x2+1351350x1x2x3+1801800x1x2x3+200200x1x3 3 3 4 5 2 4 2 +900900x1x2x4+900900x1x2x3x4+45045x1x4+270270x1x2x5 5 5 6 +72072x1x3x5+36036x1x2x6+1716x1x7. It should be noted that our results for B8,7 B9,7 and B10,7 are in full agreement with those recorded in the work (for instance) of Charalambides [4, p. 417]. New identities for the partial Bell polynomials 5 One further illustration of an application of (1.5) is the following (presumably) new explicit formula k 1 n−1 α1−1 αk−1−1 n α1 αk−1 S(n,k+1) = ··· ··· (3.1) (k+1)! αX1=k α2X=k−1 αXk=1 (cid:18)z α1(cid:19)(cid:18)α2(cid:19)}| (cid:18) αk (cid:19){ k (n ≥ k+1,k = 1,2,...) | {z } for the Stirling numbers of the second kind S(n,k) defined by means of (see [1, Chapter 5]) k 1 k S(n,k) = (−1)k−α αn, (3.2) k! α α=0 (cid:18) (cid:19) X which is an immediate consequence of the relationship S(n,k) = B (1,...,1) [1, n,k p. 135, Eq. (3g)]. Moreover, for given k, it is easy to sum the multiple sum (3.1) by repeated use of the familiar result (1+x)n = n n xk, so that we have: k=0 k 1 S(n,2) = 2n −2 = 2n−1−1, P (cid:0) (cid:1) 2 1(cid:16) (cid:17) S(n,3) = 3n −3·2n +3 , 6 1(cid:16) (cid:17) S(n,4) = 4n −4·3n+3·2n+1−4 , 24 1(cid:16) (cid:17) S(n,5) = 5n −5·4n +10·3n −10·2n +5 , 120 1 (cid:16) (cid:17) S(n,6) = 6n −6·5n +15·4n −20·3n +15·2n −6 , 720 (cid:16) (cid:17) andtheseexpressionsagreefullywiththosewhichareobtained byusingthedefining relation (3.2). Acknowledgements TheauthoracknowledgesfinancialsupportfromMinistryofScienceoftheRepublicofSerbia under Research Projects 144004 and 142025. References [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Dordrecht,1974. [2] M. Hazewinkel(Ed.), Encyclopedia of Mathematics, Supplement I, KluwerAcademic Publish- ers, Dordrehct,1997. [3] E.T. Bell, Exponentialpolynomials, Ann. Math. 35 (1934), 258–277. [4] C.A.Charalambides,EnumerativeCombinatorics,ChapmanandHall/CRC,BocaRaton,2002. [5] R. Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Scientific, Singapore, 2001. [6] H.W. Gould, Coefficient identities for powers of Taylor and Dirichlet series, Amer. Math. Monthly 81 (1974), 3–14. [7] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, 2007.