The defining characteristic of the Bernoulli numbers operationally is that they are the basis of the unique Appell sequence, the Bernoulli polynomials, that “translate” simply under the generalized binomial transform (Appell property) and satisfy (for an analytic function, such as the exponential or logarithm, when convergent, or order by order for a formal power series) the umbral relation
where is the derivative w.r.t. . This then determines their umbral compositional inverse, the “reciprocal polynomials”, based on the reciprocal integers. From the derivative and translation property, the Euler-Maclaurin results follow easily as well as from the reciprocal e.g.f.s in operator form of the pair of Appell polynomials–that the e.g.f.s are reciprocals of each other, the operators are inverses, and the polynomials are umbral compositional inverses are inextricably linked. What isn’t evident from the Euler-Maclaurin perspective are the intimate associations to Lie theory, matrix reps, and simplices of this reciprocal pair of polynomials. In the previous post, I show some of the connections among the Bernoulli polynomials, their umbral reciprocals, Lie theory, matrix reps, and the simplices.
Other connections of the Bernoullis to geometry/topology/physics through alternating permutations ([MOQ]) and hence to integrals related to volumes of certain structures can be found in these articles by N. [Elkies] and by A. [Hodges] and C. [Sukumar], related to quantum Weyl-Heisenberg type algebras. Note the connections of the analyses to trig functions and the connection to hyperbolic functions of the raising/creation ops of the Bernoulli and reciprocal polynomials below. It is in this sense (and there is far more) that the dance of the Bernoullis and the reciprocals explains their presence (to me at least) in some many domains of math and physics.
First, a review of the Appell formalism. Appell polynomial sequences are an extension of the Kronecker delta base sequence , i.e., . Translating the sequence to a higher plane through the enchanted binomials, we encounter the power basis
with lowering (destruction / annihilation) operator and raising (creation) operator , i.e., with commutator .
(So already we see shadows of the Heisenberg-Weyl Lie algebra in the iconic Appell sequence, and it’s no surprise that the probabilist’s Hermite (Appell) polynomials appear for the harmonic oscillator in quantum mechanics and the Bernoullis for other Q.M. domains.)
The exponential generating fct. for the Appell sequence is then
and an ordinary generating fct. is the formal Borel-Laplace transform of the e.g.f.
We rise to an even higher plane with the intuitive Ramanujan and use his Master Formula (the Mellin transform) to make the indices continuous
so we have the Mellin transform or a Newton-Gauss interpolator for extending (and analytically continuing to the complex domain) the base sequence. For the Kronecker base sequence, this is .
Now simply apply the formalism to the Bernoulli numbers and out pops the Bernoulli polynomials and the Hurwitz zeta function, which specializes to the Riemann zeta for , for which .
Define the umbral compositional inverse by
Then use the translation property twice to give
and setting defines the base sequence of the umbral inverse as
(Relation to Grothendieck’s axiomatic formulation of Chern class, courtesy of Wikipedia.(?))
Exponentiating helps us to readily interpret this as
The e.g.f.s of the base sequences are reciprocals of each other. This means the base sequences (and these could be almost any abelian numbers, operators, matrices, etc.) are connected by the combinatorics of surjections and permutohedra [A133314] ([A049019]), among other important implications.
Back to the Bernoullis extended to polynomials defined by
with the umbral inverse polynomials, their escorts, the elegant reciprocal integers,
The e.g.f.s morphed into operators give you the Euler-Maclaurin expansion (and more since the two e.g.f.s for the base sequence are inverse by construction, independent of their interpretation as shift operators) .For an analytic function (or term by term for a formal power series) ,
The operators are clearly an inverse pair from the umbral inverse properties and commute, so
giving (consistent with the op defn. of the Bernoulli polynomials)
Using these properties and expanding (usually with asymptotic results, see Hardy, Divergent Series),
the Euler-MacLaurin series can easily be generated as well as identities such as Faulhaber’s. Faulhaber’s formula follows simply from the umbral compositional relation, or equivalently, the operational definition of the Bernoulli polynomials acting on , using the Appell translation property and , which follows easily from the e.g.f. of the polynomials:
so summing and telescoping gives
Similarly, the same basic relations give,
Now to the raising operators, for the Bernoullis
Likewise for the umbral inverse,
and we are conjugating the basic raising op for the Kronecker base sequence. There’s more hidden here, and we can reveal it by invoking a commutator and the Pincherle derivative:
For a general pair of lowering and raising ops, the Pincherle derivative is
so we expect the Bernoullis to pop up in all of these algebras one way or another, and we have several further interesting relations (recall the e.g.f.s are reciprocals):
and so, with a simple change of sign,
which hold for general Appell sequences. For any Appell sequence, the raising op separates into , so the commutator remains invariant upon substitution of any Appell raising op for in the commutator. The Pincherle derivative is “conjugated” by the basic underlying e.g.f. of each Appell sequence. But be careful–the derivative must be taken before any further operations are done, so at this level the advantages of normal conjugation in simplifying powers of an op do not apply; however, the advantage returns in another manifestation of the op.
More specifically for the Bernoulli couple, working out the Pincherle derivative and bouncing between e.g.f.s lead to a pairing and back to our familiar Riemann zeta
( in the exponential here is meant to be shorthand for .)
So we can see how deeply entwined the reciprocals of the integers, the Riemann zeta, and the Bernoullis are with each other and important families of operator algebras. The following notes some further applications of these relations to diverse areas of mathematics.
Using the Mellin-Riemann-Ramanujan interpolation, the natural extension of the Bernoullis is the Hurwitz zeta function
which with becomes , and for the reciprocal integers,
The two are related through umbral composition and inversion, so that the pole singularities are reflected in each other. The Gauss-Newton series and umbral composition lead to the (ultimately divergent) expansion
but which gives very good results truncated to just eight terms over the range of reals –it’s capturing the dependence of zeta on the singularity, the falling factorials of , and zeta’s first three simple zeroes as a truncated approximation–ten terms captures the dependence on the next zero, where , with e.g.f. , are determined by in which , with the e.g.f. , come from the umbral composition of the Bernoulli polynomials with the Bernoulli numbers These then are specialized values of the generalized Bernoulli polynomials. (See a later entry for more details.)
The e.g.f. derivative operators associated to the Bernoulli and reciprocal polynomials contain the same amount of information as the raising and lowering operators and are encountered perhaps most frequently in relation to the Euler-Maclaurin series (but they are not necessarily the shortest or even rigorous route for solving problems). At the level of umbral composition interpreted in terms of operations on functions, the Euler-Maclaurin operations are equivalent to
Coates and Givental, “[Quantum cobordisms and formal group laws]” (pg. 15), introduce the Bernoullis through Euler-Maclaurin. In another paper, (Dunne and Schubert, “[Bernoulli numbers identities from quantum field theory and topological string theory]”), the Bernoullis slip in through differentiation (pg 3) and the digamma fct, which is another raising operator for another Appell sequence (the gamma genus) involving the Riemann zeta for positive integer values and cyle index polynomials, which has all the dressings of Chern characteristic classes (MOQ-). Recall that there is a reflection formula for the zeta fct, pg. 3 D & S and [MOQ-112062], relating the even positive integer values of zeta to the Bernoulli numbers as in the formulas for the raising ops above.
The Bernoulli and reciprocal polynomials arise often in relation to Lie theory–not surprisingly, given their connections to the exp and log and to differentiation and integration. Several applications can be found in the references below. Often the role of the reciprocal polynomials are not explicitly recognized. For example, this reference rife with the Bernoullis (thanks to [MOQ-16169] and Zoran Skoda) “[A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra]” by Durov, Meljanec, Samsarov, and Skoda gives an alternate briefer proof of their results using a dual approach (coderivations) that runs parallel to the Appell formalism above, no doubt unaware of the connection. Page 43 Eqn.37 is a g.f. for , umbrally evaluated, with and in the paper. The authors also encounter in their original approach the Fibonnaci polynomials [A011973] (pg. 16), with no remark on their identity, which are also the Pascal rows read along anti-diagonals (no doubt connected to the duality) and therefore connected to the e.g.f. for –they are also the coefficients for the characteristic polynomials of the Coxeter adjacency matrix for , related to the Chebyshev polynomials of the second kind, and to Cartan matrices, and the shifted version, well, that has exciting connections to crossing partitions, positroids, and a general Appell sequence related to compositional inversion and … . The Appell formalism should bring out sharper connections between all these structures.
The o.g.f.s associated with the umbral pair have important applications as well. The o.g.f.
where with inverse , a special Mobius transformation (which is the iconic o.g.f.), has the compositional inverse
The forms are related to colored compositions. You can read how the inverse o.g.f. is related to Eulerians, permutahedra, probablity theory, a Weierstrass elliptic function, and a formal group law, related to a generalized cohomology, through comments and references in [A008292] and [A074909]. It can also be rewritten in terms of the e.g.f.s of the Bernoullis and their umbral inverses. (The combinatorics that underlie reciprocation and compositional inversion are those of the permutahedra, associahedra, crossing partitions, and the myriad combinatoric structures related to them, so no surprise that they make an appearance in all this.)
So, we have this interplay among the Mobius transformation, reciprocation and mutiplicative inversion, umbral and regular composition, and umbral and regular compositional inversion of the logarithm and the exponential that accounts in my mind for the prevalence of the Bernoullis and pairing with the reciprocal integers, and of course the royal binomials (Pascal matrix). The Bernoullis are intimately related to differentiation and, therefore, clearly to the exponential; defines them, and the connection of the exponential operator to Lie theory, movement around a manifold, is the fundamental action translation, which is also at the heart of the Appell formalism, in both the indices and the independent variable.
is an e.g.f. for signed reverse face polynomials of the permutahedra and has the infinitesimal generator
G. Rzadkowski in “[Bernoulli numbers and solitons revisited]” explicitly shows the links between derivatives of to solutions of the Ricatti differential equation, soliton solns. of the KdV equation, and the Eulerian and Bernoulli numbers. To apply the results here, let with and note the ODEs in A145271; i.e.,
For more on the Ricatti equation, see “Lie systems: theory, generalisations, and applications” by Carinena and Lucas (http://arxiv.org/abs/1103.4166), “Dirac-Lie systems and Schwarzian equations” by Carinena, Grabowski, Lucas, and Sardon (http://arxiv.org/abs/1305.6276), and other refs below.
In addition, the comp. inversion formula [A145271] connects products of derivatives of and the refined Eulerian numbers to , which gives the face polynomials for the dual of the permutohedra. Or, apply the inversion method of [A134264] (intimately related to Appell polynomials in general and associated interpolated families of polynomials spanning the Coxeter group ) to
and you get a relation between noncrossing partitions or Dyck lattice paths weighted by the normalized Bernoullis and the face polynomials of the simplices. (Now morph all of this into totally non-negative grassmannians, positroids, binary trees, operads, and computations of characteristic classes of genera (Hirzebruch) and you have can have an exciting math weekend. Don’t forget moments, cumulants, and continued fractions.)
For relations to Hirzebruch genera / Todd class, see my other entries here.
The formulations above apply to the symmetric functions quite nicely. The generating functions for the complete homogeneous symmetric functions, the elementary symmetric functions, and the power sums, and respectively, (cf., e.g., Macdonald, “A new class of symmetric functions” and Newton Identities on Wikipedia) satisfy
so, for the raising ops of the symmetric polynomials, the analogs of the raising ops for the Bernoulli polynomials
whose n’th iterates will generate Appell polynomials in , e.g., , that when evaluated at are proportional to the complete and elementary symmetric polynomials, i.e., and . The series have the generating functions
By taking logs, the first two sets of symmetric polynomials can be related to the power sums through the cycle index partition polynomials of the symmetric group, expessed umbrally as
where demarks the level at which umbral evaluation must occur. See the Wiki article for explicit expressions of the symmetric polynomials and my MOQ on the Riemann zeta and an Appell sequence related to fractional calculus to see how info on the simplices are encoded in the numerical coefficients . The extended elementary and complete homogeneous symmetric polynomials are Appell sequences by construction and in fact are an umbral compositional inverse pair with all the nice properties of such sequences.
Note also that e.g.f.s for Appell polynomials satisfy an evolution equation determined by the raising op:
If we look at the Mellin transform for the Riemann zeta function and interpret it as an interpolation of the coefficients of the e.g.f. of the Bernoulli numbers, we can make a case for Ramanujan’s method of summation of the divergent zeta series. Make these heuristic, formal associations
Then using Riemann’s Mellin transform definition of the Riemann zeta function
Similarly, but formally,
so, from the formal definition of above, (not literally, only formally, of course, as Hardy had the breadth, keeness, and generousity of mind to recognize), and working in reverse interchanging summations, we have our Ramanujan summation of divergent series typically resulting from the formal interchanges of summations and integrations as Euler, Heaviside, Hardy, and Ramanujan put to such exemplary use.
You could also view the derivative component of the raising op of the Bernoulli polynomials as an associated series for a regularized Chern character (the finite part), using Faulhaber’s formula in the limit as tends to infinity, with
Compare with the Chern character for a sum of line bundles
The remaining notes here focus on a particular formal group law (FGL), noted in the generalized Todd class entry here, related to the Lah numbers. For other FGLs, see the other entries here on the Bernoullis. The more general Chern theory of classes seems to be related to the cycle index partition polynomials (CIP) of the symmetrc group being an Appell sequence in the indeterminate more than to specific properties of the Bernoulli polynomials; however, the Lah polynomials do appear in the discussion, and the CIP can be considered umbral generalizations of the Lah polynomials, as I show in my notes Lagrange a la Lah.
There are several topics related to the discussion on formal group laws in the Todd class entry. First, the CIP and Lah polynomials are related to enumeration of binary trees and, I believe, therefore, to binary quadratic operads. Frederic Chapoton writes about the two, and Don Rawlings has a comprehensive view of binary trees in “A binary tree decomposition space on permutation statistics”. Tree reps for special polynomials are not unique–different tree reps fit different contexts, e.g., the Lah polynomials and CIPs can be represented by weighted Cayley forests of binary (but not strictly binary) trees representing the iterated ops , which can also be used for Lagrange inversion.
Second, the falling factorials and the rising factorials are related through the signed Lah polynomials, which are in turn related to the associated Laguerre polynomials by :
These can be confirmed using the Vandermonde identities and , or through the formalism of connection coefficients presented by Mullin and Rota in “On the Foundations of Combinatorial Theory III Theory of Binomial Enumeration. ” A third way is through the operational relations
where , by definition, for any pair of operators. These are concommitant with the umbral inverse relationship where are the Bell polynomials defined operationally by or by the e.g.f. The umbral compositional inverses of the Bell polynomials are then given by the compositional inverse of , which is , through That the rising factorial and the Bell polynomials are umbral inverses follows from the inverse pair of functions:
The signed Lah polnomials themselves are a binomial Sheffer sequence that is obviuosly self-inverse under compositional inversion, as can be seen from their series and operational definitions above as well as their e.g.f. , i.e., , which can be derived from their op definition.
Combinatorial interpretations of the rising and falling factorial polynomials can be found on the OEIS and in “Set maps, umbral calculus, and the chromatic polynomial” by G. Wiseman, “Chromatic polynomials and partition systems” by C. Lenart and N. Ray, and “From sets to functions: three elementary examples” by Joni and Rota, in addition to the other Rota reference.
For diagrammatic presentations of the number arrays discussed here, see Robert Dickau’s website http://www.robertdickau.com/ .
Note the relation of the Bernoulli numbers to the zigzag numbers OEIS-000111 (cf. Wikipedia, Hodges and Sukumar references, alternating permutations.)
Papers by Franz Lehner et al. on cumulants with relations to log of the momemt generating function, non-crossing partitions, composition, multiplicative reciprocals, and Lagrange inversion:
http://arxiv.org/abs/1408.2977, Relations between cumulants in noncommutative probability
http://arxiv.org/abs/math/0110031, Cumulants, lattice paths, and orthogonal polynomials
http://arxiv.org/abs/math/0110030, Free cumulants and enumeration of connected partitions
Papers by Di Nardo et al. on cumulants and umbral calculus.
“The Magnus expansion, trees, and Knuth’s rotation correspondence” by Ebrahimi-Fard and Manchon http://arxiv.org/abs/1203.2878
“Hodge Integrals and Gromov-Witten” by Faber and Pandharipande http://arxiv.org/abs/math/9810173
“Quantum Barnes function as the partition function of the resolved conifold” by Koshkin http://arxiv.org/abs/0710.2929
“Geometry and physics” by Atiyah, Dijkgraaf, and Hitchin http://m.rsta.royalsocietypublishing.org//content/368/1914/913
“Hopf algebras in dynamical systems theory” by Carinena, Ebrahimi-Fard, Figueroa, and Gracia-Bondi http://arxiv.org/abs/math/0701010
“Functional equations and Lie algebras” by E. Petracci
“Clifford algebras and Lie groups” (2011 2009 lecture notes, 2011 version) by E. Meinrenken
“On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras” by Feher, Gabor, and Pusztai http://arxiv.org/abs/math-ph/0105047
“A note on a canonical dynamical r-matrix” by Pusztai ad Feher http://arxiv.org/abs/math/0109082
“The Kashiwara-Vergne conjecture and Drinfelds’s associators” by Alekseev and Torossian http://arxiv.org/abs/0802.4300
“On triviality of the Kashiwara-Vergne conjecture for quadratic Lie algebras” by Alekseev and Torossian http://arxiv.org/abs/0909.3743
For other refs on the Ricatti equation and linear fractional transformations see
“Hopf algebras in dynamical systems theory” by Carinena, Ebrahimi-Fard, Figueroa, Gracia-Bondi http://arxiv.org/abs/math/0701010
“Elie Cartan and geometric duality” and “An introduction to Lie groups and symplectic geometry” by Robert Bryant
“Lie algebras, representations, and analytic semigroups through dual vector fields” by P. Feinsilver http://chanoir.math.siu.edu/MATH/Merida/PDF/Merida.pdf
For the Todd operator, see “Computing the continuous discretely” by Beck and Robbins
For more on relations to the Duflo isomorphism and wheel diagrammatics see “Wheeling: A diagrammatic analogue of the Duflo isomorphism” by D. Thurston and “Differential operators and the wheels power series” by Kricker
For some discussion of Kummer congruences for reciprocal pairs of Appell sequences, see “Applications of the classical umbral calculus” by Ira Gessel