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Higher Order Bernoulli and Euler Numbers

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Higher Order Bernoulli and Euler Numbers David Vella, Skidmore College [email protected] Generating Functions and Exponential Generating Functions • Given a sequence {푎푛} we can associate to it two functions determined by power series: • Its (ordinary) generating function is ∞ 풏 푓 풙 = 풂풏풙 풏=ퟏ • Its exponential generating function is ∞ 풂 품 풙 = 풏 풙풏 풏! 풏=ퟏ Examples • The o.g.f and the e.g.f of {1,1,1,1,...} are: 1 • f(x) = 1 + 푥 + 푥2 + 푥3 + ⋯ = , and 1−푥 푥 푥2 푥3 • g(x) = 1 + + + + ⋯ = 푒푥, respectively. 1! 2! 3! The second one explains the name... Operations on the functions correspond to manipulations on the sequence. For example, adding two sequences corresponds to adding the ogf’s, while to shift the index of a sequence, we multiply the ogf by x, or differentiate the egf. Thus, the functions provide a convenient way of studying the sequences. Here are a few more famous examples: Bernoulli & Euler Numbers • The Bernoulli Numbers Bn are defined by the following egf: x Bn n x x e 1 n1 n! • The Euler Numbers En are defined by the following egf: x 2e En n Sech(x) 2x x e 1 n0 n! Catalan and Bell Numbers • The Catalan Numbers Cn are known to have the ogf: ∞ 푛 1 − 1 − 4푥 2 퐶 푥 = 퐶푛푥 = = 2푥 1 + 1 − 4푥 푛=1 • Let Sn denote the number of different ways of partitioning a set with n elements into nonempty subsets. It is called a Bell number. It is known to have the egf: ∞ 푆 푥 푛 푥푛 = 푒 푒 −1 푛! 푛=1 Higher Order Bernoulli and Euler Numbers th w • The n Bernoulli Number of order w, B n is defined for positive integer w by: w x B w n xn x e 1 n1 n! th w • The n Euler Number of order w, E n is similarly defined for positive integer w as: 푤 ∞ 2푒푥 퐸푤 = 푛 푥푛 푒2푥 + 1 푛! 푛=1 Reversing the Process • If you start with a function and ask what sequence generates it (as an ogf), the answer is given by Taylor’s theorem: 푓 푛 (0) 푐 = 푛 푛! • More generally, if we use powers of (x-a) in place of powers of x, Taylor’s theorem gives: 푓 푛 (푎) 푐 = 푛 푛! • Let us abbreviate the n’th Taylor coefficient of f about x = a by: 푓 푛 (푎) 푇 푓; 푎 = 푛 푛! A Key Question • I have mentioned that operations on the functions correspond to manipulations of the sequence. What manipulations correspond to composing the generating functions? • That is, we are asking to express 푇푛 푓 ⃘푔; 푎 in terms of the Taylor coefficients of f and of g. • In January, 2008, I published a paper entitled Explicit Formulas for Bernoulli and Euler Numbers, in the electronic journal Integers. In this paper, I answer the above question and give some applications of the answer. A Key Answer • Since 푓 ⃘푔 (푛)(푎) 푇 푓 ⃘푔; 푎 = , 푛 푛! the answer would depend on evaluating the numerator, which means extending the chain rule to nth derivatives. This was done (and published by Faà di Bruno in 1855.) I noticed (in 1994) a corollary of di Bruno’s formula which exactly answers the above question. • Francesco Faà di Bruno: THE MAIN RESULT With the above definitions, if both 푦 = 푓(푥) and 푥 = 푔(푡) have n derivatives, then so does 푦 = 푓 ⃘ 푔(푡), and 푛 풍 흅 푻 풇 ⃘품; 풂 = 푇 (푓; 푔 푎 ) 푇 (푔; 푎) 휋푖 풏 휹 흅 푙 휋 푖 흅∈푷풏 푖=1 where 푃푛 is the set of partitions of n, 푙 휋 is the length of the partition 휋, and 휋푖 is the multiplicity of i as a part of 휋. Here, 훿 휋 is the set of multiplicities {휋푖} which is itself a partition of 푙 휋 (I call it the derived partition ), and 푙 휋 is the associated 훿 휋 훿 휋 푙 휋 ! multinomial coefficient . 휋1!휋2!…휋푛! Illustration – How to use this machine • Let 푓 푥 = 푒푥 and let 푥 = 푔 푡 = 푒푡 − 1 Set 푎 = 0 and observe 푔 푎 = 0. Then: 1 푇 푓; 푔 0 = for all m and similarly, 푚 푚! 1 푇 푔; 0 = if 푖 ≥ 1. The right side becomes: 푖 푖! 푛 휋 푛! 푙 휋 1 1 푖 푛! 훿 휋 푙(휋)! 푖! 휋∈푃푛 푖=1 1 1 푛! 1 푆 = 푆 = 푛 푛! 훿 휋 ! 휋! 푛! 휋 푛! 휋∈푃푛 휋∈푃푛 Illustrations, continued • But the left hand side is 푇푛 푓 ⃘푔; 0 for the composite function 푡 푓 ⃘푔(푡) = 푒 푒 −1 So we just proved this is the egf for the sequence of Bell numbers 푆푛. This is a very short proof of a known (and famous) result. Likewise, I can provide new proofs of many combinatorial identities using this technique. Can we discover new results with this machine? Yes! Illustrations, continued ln(1+푥) Let 푓 푥 = and 푔 푡 = 푒푡 − 1. Again if 푎 = 0 then 푥 푡 푔 푎 = 푔 0 = 0. Also we have 푓 ⃘푔 푡 = which is 푒푡−1 precisely the egf of the Bernoulli numbers. In this case, our machine yields the formulas: ( ) (1 ) ( ) n Bn and: Pn 1 ( ) ( ) n (1)m Bn m !S(n, m) m1 1 m where S(n,m) is the number of ways of partitioning a set of size n into m nonempty subsets (a Stirling number of the 2nd kind.) Illustrations, continued • This was published in my 2008 paper, along with similar formulas for Euler numbers. • These formulas can be generalized in at least two ways: 1. Express the higher order Bernoulli numbers in terms of the usual Bernoulli numbers, since the egf for them is obviously a composite. Similarly for the higher order Euler numbers. This result is still unpublished (but I spoke at HRUMC a couple of years ago about it.) 2. Generalize my corollary of di Bruno’s formula to the multivariable case. (The next talk is about this in the case of multivariable Bernoulli numbers!) • For the remainder of this talk, I’d like to focus on one case where the identity I obtain from my machine is not obviously useful (or is it?) Final Example: the Catalan numbers • Recall from an earlier slide the generating function (ogf) of the Catalan numbers: ∞ 푛 1 − 1 − 4푥 2 퐶 푥 = 퐶푛푥 = = 2푥 1 + 1 − 4푥 푛=1 • This can be expressed as a composite generating function as 2 follows: Let 푓 푢 = and let 푢 = 푔 푥 = 1 − 4푥. Then 1+푢 퐶 푥 = 푓 푔 푥 . If 푎 = 0 then 푔 푎 = 1, so the derivatives and Taylor coefficients of 푓 푢 have to be evaluated at 푢 = 1. (−1)푚푚! • It is easy to check by direct calculation that 푓 푚 1 = 2푚 푓 푚 1 (−1)푚 and therefore 푇 푓; 1 = = . 푚 푚! 2푚 Catalan numbers, continued • To find 푇푖(푔; 0), we write the radical as a power and use Newton’s binomial series: 1 ∞ 1 푘 푘 2 2 푔 푥 = 1 − 4푥 = 푘=0 푘 −4 푥 . 1 2 푖 It follows that 푇푖 푔; 0 = 푖 −4 . So our formula yields: 휋푖 푛 푙 휋 −1 푙(휋) 1 퐶 = 2 −4 푖 푛 훿 휋 2푙(휋) 푖 휋∈푃푛 푖=1 However, we can simplify this because 푛 휋 푖 푖 푖휋 푛 −4 = −4 푖 = −4 푖=1 Simplifications Thus, 푛 휋푖 푙 휋 −1 푛+푙(휋) 1 퐶 = 4푛 2 푛 훿 휋 2푙(휋) 푖 휋∈푃푛 푖=1 Next, we rewrite the terms 휋푖 휋푖 1 1 1 1 1 − 1 − 2 … − 푖 + 1 2 = 2 2 2 2 푖 푖! When we take the product of these terms over i, the 푛 휋푖 denominator becomes 푖=1 푖! , the product of the factorials of all the parts of 휋, which I abbreviate as 휋!. Observe that 1 푛 = 푛! 휋! 휋 Looking more carefully at the numerator, we obtain: Simplifications, continued 휋 휋 1 1 1 1 푖 1 1 3 2푖 − 3 푖 − 1 − 2 … − 푖 + 1 = − − … − 2 2 2 2 2 2 2 2 휋 We can factor out a power of 2 in the denominator, namely 2푖 푖 = 2푖휋푖, and since every factor except the first is negative, we can also factor out a 휋 power of -1, namely (−1)푖−1 푖 = −1 푖휋푖−휋푖. Now when we take the product over i, this means we factor out the following: in the denominator, 2푖휋푖 = 2 푖휋푖 = 2푛 (which cancels part of the 4푛 outside the sum), and in the numerator, (−1)푖휋푖−휋푖 = (−1) 푖휋푖− 휋푖= (−1)푛−푙(휋). Combined with the (−1)푛+푙(휋) term in front of the product, this becomes (−1)2푛= 1. In other words, all the negatives cancel out! Finally, what remains in the square brackets is a product of odd integers, which we abbreviate with the double factorial notation (with the convention that (-1)!! = 1). The entire thing simplifies to: New formula for the Catalan numbers! • We have proved: 푛 푛 2 푛 푙 휋 1 휋 퐶 = 2푖 − 3 ‼ 푖 푛 푛! 휋 훿 휋 2푙(휋) 휋∈푃푛 푖=1 Let’s illustrate this with n = 4. We need the table: 흅 흅ퟏ 흅ퟐ 흅ퟑ 흅ퟒ 풍(흅) 휹(흅) [4] 0 0 0 1 1 [1] [3,1] 1 0 1 0 2 [12] [22] 0 2 0 0 2 [2] [2, 12] 2 1 0 0 3 [2,1] 14 4 0 0 0 4 [4] Catalan example (n = 4) • Our formula becomes: 4 4 2 4 푙 휋 1 휋 퐶 = 2푖 − 3 ‼ 푖 4 4! 휋 훿 휋 2푙(휋) 휋∈푃4 푖=1 16 4 1 1 4 2 1 4 2 1 = (5‼)1 + (3‼)1 + (1‼)2 24 4 1 2 3 1 1 1 22 2 2 2 22 16 4 3 1 4 4 1 + (1‼)1 + ((−1)‼)4 24 2 1 1 2 1 23 1 1 1 1 4 24 16 1 1 1 = 1 ∙ 1 ∙ ∙ 5 ∙ 3 ∙ 1 + 4 ⋅ 2 ⋅ ⋅ 3 ⋅ 1 + 6 ⋅ 1 ⋅ ⋅ 12 24 2 4 4 16 1 1 + 12 ∙ 3 ∙ ⋅ 1 + 24 ⋅ 1 ⋅ ⋅ 1 24 8 16 Catalan example (n = 4) = 5 + 4 + 1 + 3 + 1 = 14 This is the correct value as 1 퐶 = 8 = 14 4 5 4 Of course it appears as if my formula is kind of useless since it is so inefficient! Or maybe not....
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