AAn introduction to theBernoulli function
Peter H. N. Luschny
Abstract.
The
Bernoulli function B( s, v ) = − s ζ (1 − s, v ) inter-polates the Bernoulli numbers but can be introduced independentlyof the zeta function. The point of departure is a modification ofthe Stieltjes constants based on an integral representation givenby J. Jansen. The functional equation of B( s, v ) and its relationto the Riemann ζ and ξ function is explored. Classical results ofHadamard, Worpitzky, and Hasse are recast in terms of B( s, v ) . The extended Bernoulli function defines the Bernoulli numbers forodd indices harmonizing with rational numbers studied by Eulerin 1735 and which are the bridge to the
Euler and André numbers .Interpolating functions for both the signed and the unsigned caseare given. The
Swiss knife polynomials let the integer sequences ofthe Euler–Bernoulli family calculate easily.
Synopsis Index Plots Contents Prologue: extension by interpolation
The question
André Weil recounts the origin of the gamma function in hishistorical exposition of number theory [66, p. 275]:
Ever since his early days in Petersburg Euler had beeninterested in the interpolation of functions and formulasgiven at first only for integral values of the argument;that is how he had created the theory of the gammafunction.
The three hundred year success story of Euler’s gamma functionshows how fruitful this question is. And that the usefulness of suchan investigation is not limited by the fact that there are infinitelymany ways to interpolate a sequence of numbers.The question we will explore in this essay is: how can theBernoulli numbers be interpolated most meaningfully?1 a r X i v : . [ m a t h . HO ] S e p he method The Bernoulli numbers had been known for some time at thebeginning of the 18th century and used in the (now Euler–Maclaurincalled) summation formula in analysis, first without realizing thatthese numbers are the same in each case.Then, in 1755, Euler baptized these numbers
Bernoulli numbers in his
Institutiones calculi differentialis (following the lead of deMoivre). After that, things changed, as Edward Sandifer [57] tells: ... for once the Bernoulli numbers had a name, theirdiverse occurrences could be recognized, organized, ma-nipulated and understood. Having a name, they madesense.
The function we are going to talk about is not new. However,it is not treated as a function in its own right and with own name.So we will give the beast a name. We will call the interpolatingfunction the
Bernoulli function . If it is true, as Barry Mazur[50] explains, that the Bernoulli numbers “act as a unifying force,holding together seemingly disparate fields of mathematics,” thenthis should be reflected all the more in this function.
What to expect
This note is best read as an annotated formula collection, all theproofs are in the references.The Bernoulli function and the Riemann zeta function are soprofoundly interwoven that one’s properties can easily be derivedfrom the properties of the other. For instance all the questionsRiemann associated with the zeta function can also be discussedwith the Bernoulli function.The introduction of the Bernoulli function leads to greatercoherence. It can advise ‘generatingfunctionologists’ [67] whichnumbers to hang up on their clothesline (see also [41]) and in manycases leads to simpler and more natural representations.Seemingly the first to treat the Bernoulli function very con-cretely in our sense was J. Jensen [32], who gave an integral formulafor the Bernoulli function of remarkable simplicity. Except for refer-encing Cauchy’s theorem, he did not develop the proof. The proofis worked out in I. Blagouchine and F. Johansson [34].2 ynopsis eq.no. B( s, v ) = 2 π (cid:90) ∞−∞ ( v − / iz ) s (e − πz + e πz ) dz Jensen formula → B( s ) = B( s,
1) = − s ζ (1 − s ) Bernoulli function → B c ( s ) = B ( s, / Central Bernoulli function → B ∗ ( s ) = B( s )(1 − s ) Alternating Bernoulli function → B n = B( n ) Bernoulli numbers → G( s, v ) = 2 s (cid:18) B (cid:16) s, v (cid:17) − B (cid:18) s, v + 12 (cid:19)(cid:19) Genocchi function → G( s ) = G( s,
1) = 2 s (B c ( s ) − B( s )) Genocchi function → E( s, v ) = − G( s + 1 , v ) s + 1 Generalized Euler function → E( s ) = E( s, Euler function → E c ( s ) = 2 s E( s, ) Central Euler function → E n = E c ( n ) Euler numbers → (cid:101) ζ ( s ) = ζ ( s ) + ζ (cid:0) s, (cid:1) − ζ (cid:0) s, (cid:1) s − Extended zeta function → B ( s ) = − s (cid:101) ζ (1 − s ) Extended Bernoulli function → E ( s ) = 4 s +1 − s +1 ( s + 1)! B ( s + 1) Extended Euler function → A ( s ) = s ! E ( s ) = (2 s +1 − s +1 ) (cid:101) ζ ( − s ) André function → A (cid:63) ( s ) = i ( i s Li − s ( − i ) − ( − i ) s Li − s ( i )) Unsigned André function → B (cid:63) ( s ) = s A (cid:63) ( s − s − s Unsigned extended Bernoulli function → ξ ( s ) = B( s ) σ ! π σ with σ = (1 − s ) / Riemann ξ function → γ = − B (cid:48) (0) (Euler), γ n − = − n B ( n ) (0) Stieltjes constants → he zeta and the Bernoulli function in the complex plane. The Stieltjes constants
The generalized Euler constants, also called
Stieltjes constants ,are the real numbers γ n defined by the Laurent series in a neigh-borhood of s = 1 of the Riemann zeta function ([56], [1]), ζ ( s ) = 1 s − ∞ (cid:88) n =0 ( − n n ! γ n ( s − n , ( s (cid:54) = 1) . (1)As a special case they include the Euler constant γ = γ ( ≈ ) .An extensive discussion with many historical notes can be found in[13]. For the numerical values consult [40].I. V. Blagouchine [12] following J. Franel [23] shows that γ n = − πn + 1 + ∞ (cid:90) −∞ log (cid:0) + iz (cid:1) n +1 (e − π z + e π z ) dz. (2)Recently it has been observed that the integral representation (2)can be employed in a particularly efficient way to numericallyapproximate the Stieltjes constants with prescribed precision [34].4 he Bernoulli constants The
Bernoulli constants β n are defined for n ≥ as β n = 2 π + ∞ (cid:90) −∞ log (cid:0) + iz (cid:1) n (e − π z + e π z ) dz. (3)We write f ( x ) n for ( f ( x )) n and take the principal value for thelogarithm implicit in the exponential, here and in all later similarformulas.The Bernoulli function is defined as B( s ) = ∞ (cid:88) n =0 β n s n n ! . (4)Note that β = 1 , thus in particular B(0) = 1 . From (2) and (3)we see that γ n = − β n +1 / ( n + 1) . Thus we get: B( s ) = ∞ (cid:88) n =0 β n s n n != 1 + ∞ (cid:88) n =1 β n s n n != 1 + ∞ (cid:88) n =0 β n +1 s n +1 ( n + 1)!= 1 − ∞ (cid:88) n =0 γ n s n +1 n != 1 − s ∞ (cid:88) n =0 γ n s n n != − sζ (1 − s ) . (5)Here the singularity of − sζ (1 − s ) at s = 0 is removed by B(0) = 1 .So B( s ) is an entire function with its defining series convergingeverywhere in C . The Bernoulli numbers
We define the
Bernoulli numbers as the values of the Bernoulli5 . − . − . . . . .
81 1 2 3 4 − γ The Bernoulli constants seen as values of a real function. function at the nonnegative integers. According to (4) this means B n = B( n ) = ∞ (cid:88) j =0 β j n j j ! . (6)Since for n > an odd integer − nζ (1 − n ) = 0 , the Bernoullinumbers vanish at these integers and B n = 1 − n (cid:80) ∞ j =0 γ j n j j ! implies ∞ (cid:88) j =0 γ j n j j ! = 1 n and ∞ (cid:88) j =0 β j n j j ! = 0 ( n > odd ) . (7) The expansion of the Bernoulli function
The Bernoulli function B( s ) = − sζ (1 − s ) can be expanded byusing the generalized Euler–Stieltjes constants B( s ) = 1 − γs − γ s − γ s − γ s . . . , (8)or in its more natural form using the Bernoulli constants B( s ) = 1 + β s + β s + β s + β s . . . . (9)6 β ( r ) − -1.0967919991262275651322398023421657187190 . . . − / . . . / . . . -0.5772156649015328606065120900824024310421 . . . / -0.4131520868458801199329318166967102536980 . . . . . . / . . . . . . / -0.0845272473711484887663180676975841853310 . . . -0.0082153376812133834646401861710135371428 . . . The Bernoulli constants for some rational r . Although we will always refer to the well-known properties ofthe zeta function when using (5), our definition of the Bernoullifunction and the Bernoulli numbers only depends on (3) and (4).The index n in (3) is not restricted to integer values. Forillustration the function β r is plotted in the figure above, where theindex r of β is understood to be a real number. The table abovedisplays some numerical values of Bernoulli constants. Integral formulas for the Bernoulli constants
Let us come back to the definition of β n as given in (3). Theappearance of the imaginary unit forces complex integration; onthe other hand, only the real part of the result is used. Fortunately,the definition can be simplified such that the computation stays inthe realm of reals provided n is a nonnegative integer.Using the symmetry of the integrand with respect to the y -axisand (e − π z + e π z ) = 4 cosh( πz ) we get from the definition (3) β n = π (cid:90) ∞ Re (cid:0) log( + iz ) n (cid:1) cosh( πz ) dz . (10)For the numerator of the integrand we set for n ≥ σ n ( z ) = Re(log( + iz ) n ) . (11)7 . . . y − − . . x log(2) / f ( z ) = − log(( z + ) ) sech( πz ) with (cid:82) ∞−∞ f ( z ) dz = γπ . By induction we see that σ n ( z ) = (cid:98) n/ (cid:99) (cid:88) k =0 ( − k (cid:18) n k (cid:19) a ( z ) n − k b ( z ) k , (12)where a ( z ) = log( z + ) / and b ( z ) = arctan(2 z ) .Therefore the constants can be computed by real integrals, β n = π (cid:90) ∞ σ n ( z )cosh( πz ) dz . (13)Similarly for the Stieltjes constants, γ n = − πn + 1 (cid:90) ∞ σ n +1 ( z )cosh( πz ) dz . (14) Some special integral formulas
Formula (14) reads for n = 0 γ = − π (cid:90) ∞ log( z + )2 cosh( πz ) dz. (15)8his follows since for real zσ ( z ) = Re log (cid:18)
12 + iz (cid:19) = log (cid:18) (cid:112) z + 1 (cid:19) = 12 log (cid:18) z + 14 (cid:19) . Using the symmetry of the integrand with respect to the y -axisthis can be rephrased as: Euler’s gamma is π times the integral of − log(( z + ) ) sech( πz ) over the real line (see the figure above).The constant γ/π is A301813 in the
OEIS .Using the abbreviations a = log( z + ) / , b = arctan(2 z ) and c = cosh( πz ) the first few Bernoulli constants are by (12): β = π (cid:90) ∞ ac dz , (16) β = π (cid:90) ∞ a − b c dz , (17) β = π (cid:90) ∞ a − ab c dz , (18) β = π (cid:90) ∞ a − a b + b c dz , (19) β = π (cid:90) ∞ a − a b + 5 ab c dz . (20) Generalized Stieltjes and Bernoulli constants
We recall that γ n /n ! is the coefficient of (1 − s ) n in the Laurentexpansion of ζ ( s ) about s = 1 and γ n ( v ) /n ! is the coefficient of (1 − s ) n in the Laurent expansion of ζ ( s, v ) about s = 1 . In otherwords, with the generalized Stieltjes constants γ n ( v ) we have the Hurwitz zeta function ζ ( s, v ) in the form ζ ( s, v ) = 1 s − ∞ (cid:88) n =0 ( − n n ! γ n ( v )( s − n , ( s (cid:54) = 1) . (21)The generalized Stieltjes constants may be computed for n ≥ and Re( v ) > by an extension of the integral representation (2), see[34, formula 2]. γ n ( v ) = − π n + 1) (cid:90) ∞−∞ log (cid:0) v − + iz (cid:1) n +1 cosh( πx ) dz. (22)9he generalized Bernoulli constants are defined as β s ( v ) = 2 π (cid:90) + ∞−∞ log (cid:0) v − + ix (cid:1) s (e − πx + e πx ) dx. (23)Note that β ( v ) = 1 for all v and β n ( v ) = − nγ n − ( v ) for n ≥ . The generalized Bernoulli function
Next we introduce the generalized Bernoulli function B( s, v ) , which is the analog of the Hurwitz zeta function. The new parameter v can be any complex number which is not a non-positive integer.The generalized Bernoulli function is defined as B( s, v ) = ∞ (cid:88) n =0 β n ( v ) s n n ! . (24)For v = 1 this is the ordinary Bernoulli function (4). Using theidentities from the last section we get B( s, v ) = 1 − s ∞ (cid:88) n =0 γ n ( v ) s n n ! . (25)Thus the generalized Bernoulli function can be represented by B( s, v ) = − s ζ (1 − s, v ) , ( s (cid:54) = 1) . (26)This also embeds the Bernoulli polynomials as B n ( x ) = B( n, x ) ( n ≥ , n integer ) . (27)This follows from (26) (see for instance [4, Th. 12.13]) and the factthat B(0 , x ) = 1 . Integral formulas for the Bernoulli function
The integral formulas for the Bernoulli constants can be trans-ferred to the Bernoulli function itself. In the first step we reproducea formula by J. L. Jensen [32], which he gave in a reply to E. Cesàro10n the
L’Intermédiaire des mathématiciens . ( s − ζ ( s ) = 2 π (cid:90) ∞−∞ ( + iy ) − s (e πy + e − πy ) dy. (28)Jensen comments: "... [this formula] is remarkable because of its simplicityand can easily be demonstrated with the help of Cauchy’stheorem." How Jensen actually computed ( s − ζ ( s ) is unclear, since theformula for the coefficients c v , which he states, rapidly diverges.This was observed by V. Kotěšovec (personal communication).In a numerical example Jensen uses the Bernoulli constantsin the form c v = ( − v β v /v ! . Applied to the Bernoulli function,
Jensen’s formula is written as B( s ) = 2 π (cid:90) ∞−∞ ( + iz ) s (e πz + e − πz ) dz. (29)This formula can be seen as a special case of the first formula intheorem 1 in Johansson and Blagouchine [34], [61, p. 92], [60].P. Hadjicostas [26] remarks that from this theorem also thecorresponding representation for the generalized Bernoulli functioncan be derived:For all s ∈ C and v ∈ C with Re( v ) ≥ / s, v ) = 2 π (cid:90) ∞−∞ ( v − + iz ) s (e − πz + e πz ) dz. (30) The Hurwitz–Bernoulli function
The
Hurwitz–Bernoulli function is defined as H( s, v ) = e − iπs/ L( s, v ) + e iπs/ L( s, − v ) , (31) L( s, v ) = − s !(2 π ) s Li s (e πiv ) . (32)Here Li s ( v ) denotes the polylogarithm. The proposition that B s ( v ) = B( s, v ) = H( s, v ) , for ≤ v ≤ and s > , (33)11 he Hurwitz–Bernoulli functions with s = 2 + k/ , (0 ≤ k ≤ , deform B ( x ) into B ( x ) . goes back to Adolf Hurwitz. In the corresponding case for the zetafunction (33) is known as the Hurwitz formula [4, p. 71].With the Hurwitz–Bernoulli function the Bernoulli polynomialscan be continuously deformed into each other (see the figure above).
The central Bernoulli function
Setting v = 1 in (33) the Hurwitz–Bernoulli function simplifies to B s (1) = − s ! Li s (1) cos( sπ/ / (2 π ) s , ( s > . (34)For s > one can replace the polylogarithm by the zeta functionand then apply the functional equation of the zeta function to get B s (1) = − sζ (1 − s ) , ( s > . Thus the Bernoulli function is a vertical section of the Hurwitz–Bernoulli function, B( s ) = B s (1) , similarly as the Bernoulli numbersare special cases of the Bernoulli polynomials, B n = B n (1) .Setting v = 1 / in the Hurwitz–Bernoulli function leads to asecond noteworthy case. Then (33) reduces to B s (1 /
2) = − s ! Li s ( −
1) cos( sπ/ / (2 π ) s , ( s > . (35)12 . . . . .
81 2 4 6 8 10 B ( s ) B c ( s ) The Bernoulli function and the central Bernoulli function
For s > we can replace the polylogarithm by the negated alter-nating zeta function. We call B c ( s ) = B s (1 / the central Bernoullifunction . B c ( s ) has the same trivial zeros as the Bernoulli function,plus a zero at the point s = 1 . (See the plot above.)An integral representation for the central Bernoulli functionfollows from (30). For all s ∈ C B c ( s ) = 2 π (cid:90) ∞−∞ ( iz ) s (e − πz + e πz ) dz. (36) The central Bernoulli polynomials
The central Bernoulli numbers are defined as B cn = 2 n B n (1 / . (37)The first few are, for n ≥ : , , − , , , , − , , , , − , , , . . . . Unsurprisingly Leonhard Euler in 1755 in his
Institutiones alsocalculated some central Bernoulli numbers (Opera Omnia, Ser. 1,Vol. 10, p. 351). 13 x − + x − x + x − x + x x − x + x − + 7 x − x + x − x + x − x + x − x + x − x + x x − x + x − x + x The central Bernoulli polynomials B cn ( x ) .The central Bernoulli polynomials are defined, like the Bernoullipolynomials, as an Appell sequence. B cn ( x ) = n (cid:88) k =0 (cid:18) nk (cid:19) B ck x n − k . (38)The parity of n equals the parity of B cn ( x ) (the Bernoulli polyno-mials do not possess this property).Despite their systematic significance the central Bernoulli poly-nomials were not in the OEIS database at the time of writing theselines (now they are filed in
A335953 ).In the context of ‘going halves’ also the following identity isworth noting (we will come back to it later): n B n (1) = n (cid:88) k =0 (cid:18) nk (cid:19) k B k (cid:0) (cid:1) . (39) The Genocchi function
How much does the central Bernoulli function deviate from theBernoulli function? The
Genocchi function gives an answer to this(up to a scaling factor). G( s ) = 2 s (cid:0) B s ( ) − B s (1) (cid:1) . (40)From the identities (34) and (35), it follows that the Genocchi14 he derivatives of the Bernoulli function B ( n ) ( s ) , ≤ n ≤ . function can be represented, first for s > and then by analyticalcontinuation in general, as G( s ) = 2(1 − s ) B( s ) . (41)A useful property of the Genocchi function is that it takes integervalues for non-negative integer arguments. The G n = G( n ) areknown as Genocchi numbers ( A226158 ).The
Genocchi polynomials are defined as G n ( x ) = 2 n (cid:18) B n (cid:16) x (cid:17) − B n (cid:18) x + 12 (cid:19)(cid:19) . (42)They describe the difference between the Bernoulli polynomialsand the central Bernoulli polynomials, up to a scaling factor. Theinteger coefficients of these polynomials are recorded as A333303 inthe
OEIS . The Genocchi function is closely related to the alternatingBernoulli function as we will see later.
Derivatives of the Bernoulli function
The Bernoulli constants are related to the derivatives of theBernoulli function. With the Riemann zeta function we have B ( n ) ( s ) = ( − n (cid:16) nζ ( n − (1 − s ) − sζ ( n ) (1 − s ) (cid:17) . (43)Here B ( n ) ( s ) denotes the n -th derivative of the Bernoulli function.15 . . . .
751 2 4 6 8 10 12 B ( s ) = − sζ (1 − s ) − B ( s ) = − ζ (1 − s ) + xζ (1 − s ) γ O − B (cid:48) ( s ) hits Euler’s γ at s = 0 , red points are Bernoulli numbers. Taking lim s → on the right hand side of (43) we get B ( n ) (0) = β n = − n γ n − ( n ≥ . (44)The n -th derivative of the Hurwitz ζ -function with respect to s is ζ ( n ) ( s, v ) = ∂ n ∂s n ζ ( s, v ) (45)and the n -th derivative of the generalized Bernoulli function is B ( n ) ( s, v ) = ( − n ( n ζ ( n − (1 − s, v ) − s ζ ( n ) (1 − s, v )) , (46)using the limiting value if s = 0 . Rewriting (46) we have for n ≥ γ n ( v )Γ( v ) = ( − n lim s → (cid:16) n ζ ( n − (1 − s, v ) − s ζ ( n ) (1 − s, v ) (cid:17) . (47)Comparing this with the representation of the Stieltjes constantswe see that γ n (1) = γ n and that relation (44) generalizes to B ( n ) (0 , n ) = − n γ n − ( n ) ( n ≥ . (48)The plot above summarizes much of what has been said: it showsthe Bernoulli function together with the Bernoulli numbers and the16dditive inverse of the derivative of the Bernoulli function whichhits Euler’s γ at the origin.Entire books [28] have been written about the emergence ofEuler’s gamma in number theory. The identity − B (cid:48) (0) = γ is oneof the beautiful places where this manifests. The logarithmic derivative
The logarithmic derivative of a function F will be denoted by L F( s ) = F (cid:48) ( s )F( s ) . In particular we will write L B( s ) , L ζ ( s ) and L Γ( s ) for the loga-rithmic derivative of the Bernoulli function, the ζ function and the Γ function (also known as digamma function ψ ).We will also use the notation ρ ( s ) for the function ρ ( s ) = 1 s + π s ) − log(2 π ) . In terms of the zeta function L B( s ) can also be written L B( s ) = 1 s − L ζ (1 − s ) . (49)The case s = is particularly interesting. It is known that thetruth of the proposition L B (cid:18) (cid:19) + 2 = − π (cid:90) ∞ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( + it ) ζ ( ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dtt (50)is equivalent to the Riemann hypothesis (see [30, Th. 7.26]). Notethat L B( ) = 2 − π/ − γ/ − log(8 π ) / , (cf. also A335263 ).From (49) we can infer, by well known relations (see [30]), L B( s ) = L Γ( s ) + L ζ ( s ) + ρ ( s ) . (51)Expressing the ζ function by the Euler product, we can derivefrom (51) and the von Mangoldt’s function Λ for Re( s ) > , ∞ (cid:88) n =1 Λ( n ) n s = L Γ( s ) − L B( s ) + ρ ( s ) . (52)17 ernoulli cumulants The series expansion of L B( s ) = (cid:80) ∞ n =1 b n s n − at s = 0 starts L B( s ) = β + ( β − β ) s + ( β − β β + 2 β ) s /
2+ ( β − β − β β + 12 β β − β ) s / O ( s ) . The coefficients b n are given by b n = [ s n ] log (cid:18) ∞ (cid:88) n =0 β n s n n ! (cid:19) . (53)In other words, the coefficients are the logarithmic polynomials generated by the Bernoulli constants (Comtet [9], p.140). Thesepolynomials may be called Bernoulli cumulants , following a similarnaming by A. Voros [65, 3.16].The numerical values appearing in this expansion, listed as anirregular triangle, are
A263634 , row-reversed: [0] 1[1] 1, -1[2] 1, -3, 2[3] 1, [- 3, -4], 12, -6[4] 1, [-10, -5], [30, 20], -60, 24[5] 1, [-10, -15, -6], [30, 120, 30], [-270, -120], 360, -120
Worpitzky numbers
The numbers in the above triangle are refinements (indicated bythe square brackets) of the signed
Worpitzky numbers W( n, k ) ([69],[64], A163626 , A028246 ). W( n, k ) = ( − k k ! (cid:26) n + 1 k + 1 (cid:27) . (54)Here (cid:8) nk (cid:9) denotes the Stirling set numbers. Generalizations basedon Joffe’s central differences of zero are A318259 and
A318260 .The
Worpitzky transform maps a sequence a , a , a , . . . to asequence b , b , b , . . . ,b n = n (cid:88) k =0 W( n, k ) a k . (55)18f a has ordinary generating function a ( x ) , then b has exponentialgenerating function a (1 − e x )e x . Merlini et al. [49] call the transformthe Akiyama–Tanigawa transformation, in the
OEIS also the termBernoulli–Stirling transform is used.Julius Worpitzky proved in 1883: if we choose a k = k +1 andapply transform (55), the resulting sequence gives the Bernoullinumbers. This approach can be generalized. The generalized Worpitzky transform
The generalized Worpitzky transform (56) maps an integer se-quence a , a , a , . . . to a sequence of polynomials W m ( a ) , m ≥ .More formally: W : Z N → Z [ x ] N , a ∈ Z N and W( a ) is a sequenceof polynomials, the m -th term of which is the polynomial W m ( a ) . W m ( a ) = m (cid:88) n =0 ( − n (cid:18) mn (cid:19) x m − n n (cid:88) k =0 W( n, k ) a k . (56)Here the inner sum is the Worpitzky transform (55) of a . The firstfew polynomials are: a a x − ( a − a ) a x − ( a − a ) x + ( a − + ) a x − ( a − a ) x + ( a − + ) x − ( a − + − ) a x − ( a − a ) x + ( a − + ) x − ( a − + − ) x + ( a − + − + ) The definition (56) can be rewritten as W m ( a ) = m (cid:88) n =0 a n n (cid:88) k =0 ( − k (cid:18) nk (cid:19) ( x − k − m . (57)As the reader probably anticipated, we get the Bernoulli poly-nomials if we set a n = 1 / ( n + 1) . Evaluating at x = 1 we arrive ata well known representation of the Bernoulli numbers: B m = ( − m m (cid:88) n =0 n + 1 n (cid:88) k =0 ( − k (cid:18) nk (cid:19) k m , m ≥ . (58)19 he Hasse representation Enter Helmut Hasse [27] 1930, who takes the next step in thedevelopment of formula (58) and proves:
Theorem
The infinite series B( s, v ) = ∞ (cid:88) n =0 n + 1 n (cid:88) k =0 ( − k (cid:18) nk (cid:19) ( k + v ) s (59)converges for all complex s and represents the entire function − sζ (1 − s, v ) , the Bernoulli function. Corollary
The Bernoulli constants β s ( v ) may be given bythe infinite series β s ( v ) = ∞ (cid:88) n =0 n + 1 n (cid:88) k =0 ( − k (cid:18) nk (cid:19) ln( k + v ) s . (60)For proofs see [10], for variants and historical notes regarding thetheorem see [14]. The constant τ and the function τ ( s ) As the next topic we shall consider the functional equation ofthe Bernoulli function, which generalizes a formula of Euler whichKnuth et al. ([24], eq. 6.89) call almost miraculous .But before we do this let us introduce yet another function andquote a remark from Terence Tao [63].
It may be that πi is an even more fundamental constantthan π or π . It is, after all, the generator of log(1) .The fact that so many formulas involving π n depend onthe parity of n is another clue in this regard. Taking up this remark we will use the notation τ = 2 πi and thefunction τ ( s ) = τ − s + ( − τ ) − s . (61)We are using the principal branch of the logarithm when takingpowers of τ . 20 he functional equation This allows us to write the functional equation of the Riemannzeta function [56] as the product of three functions, ζ (1 − s ) = ζ ( s ) τ ( s ) Γ( s ) ( s ∈ C \ { , } ) . (62)Using (62) and B( s ) = − sζ (1 − s ) we get the representation B( s ) = − ζ ( s ) τ ( s ) s ! . (63)From B(1 − s ) = ( s − ζ ( s ) we obtain a self-referential representa-tion of the Bernoulli function, the functional equation B( s ) = B(1 − s )1 − s τ ( s ) s ! . (64)This functional equation has also a symmetric variant , which meansthat the left side of (65) is unchanged by the substitution s ← − s . B(1 − s ) (cid:16) s (cid:17) ! π − s/ = B( s ) (cid:18) − s (cid:19) ! π − (1 − s ) / . (65) Representation by the Riemann ξ function The right (or the left) side of (65) turns out to be the
Riemann ξ function , ξ ( s ) = (cid:16) s (cid:17) ! π − s/ ( s − ζ ( s ) . (66)For a discussion of this function see for instance Edwards [15]. Ournotation follows Landau, as it is usual nowadays.Thus we get a second representation of the Bernoulli functionin terms of a Riemann function: B( s ) = π (1 − s ) / ((1 − s ) / ξ ( s ) . (67)By the functional equation of ξ , ξ ( s ) = ξ (1 − s ) , we also get B(1 − s ) = π s/ ( s/ ξ ( s ) . (68)21 . . .
52 3 6 9 ξ ( s ) B ( s ) /ξ ( s ) The Hadamard decomposition of the Bernoulli function.
This is a good opportunity to check the value of B( − . B( −
1) = π ξ ( −
1) = π π ξ (2) . (69)In this row of identities the names Bernoulli, Euler (solving theBasel problem in 1734) and Riemann join together. The Hadamard decomposition
We denote Hadamard’s infinite product over the zeros of ζ ( s ) by H ζ ( s ) = 12 (cid:89) Im ρ> (cid:18)(cid:18) − sρ (cid:19) (cid:18) − s − ρ (cid:19)(cid:19) . (70)The product runs over the zeros with Im( ρ ) > . The absoluteconvergence of the product is guaranteed as the terms are taken inpairs ( ρ, − ρ ) . Hadamard’s infinite product expansion of ζ ( s ) is ζ ( s ) = π s/ ( s/ ζ ( s ) s − , s / ∈ { , − , − , . . . } . (71)Applying Jensen’s formula (28) leads to the representation H ζ ( s ) = 2( s/ π s/ − (cid:90) ∞−∞ ( + ix ) − s (e πx + e − πx ) dx. (72)22ince the zeros of ζ ( s ) and B( s ) are identical in the critical stripby (63) this representation carries directly over to the Bernoullicase. Writing σ = (1 − s ) / we get B( s ) = π σ σ ! H ζ ( s ) , s / ∈ { , , , . . . } . (73)This is the Hadamard decomposition of the Bernoulli function .The zeros of B( s ) with Im( ρ ) = 0 are at , , , . . . (making theBernoulli numbers vanish at these indices), due to the factorial termin the denominator. This representation separates the nontrivialzeros on the critical line from the trivial zeros on the real axes.(See the plot above and for complex s the appendix).Here we can see another reason why B = . The oscillatingfactor has the value π = 1 and the Hadamard factor has the value H ζ (1) = − ζ (0) · . The Bernoulli value follows from ζ (0) = − .If we compare the identities (67) and (73) , we get as a corollary ξ = H ζ . This is precisely the proposition that Hadamard proves inhis 1893 paper [25].
The alternating Bernoulli function
The alternating Bernoulli function is defined as B ∗ ( s ) = B( s ) (1 − s ) . (74)We can express the alternating Bernoulli function in terms of thezeta function using identity (63) as B ∗ ( s ) = s ! ζ ( s ) τ ( s ) (2 s − . (75)The alternating Bernoulli numbers are the values of the alternatingBernoulli function at the nonnegative integers, B ∗ n = B ∗ ( n ) . (76)Like the Bernoulli numbers the alternating Bernoulli numbersare rational numbers. Reduced to lowest terms they have thedenominator , B ∗ = 0 and B ∗ = − . In the form G n = 2 B ∗ n theyare the Genocchi numbers introduced above as the values of theGenocchi function at nonnegative integers.23he alternating Riemann zeta function , also known as the Dirichlet eta function , is defined as ζ ∗ ( s ) = ζ ( s )(1 − − s ) , ( s (cid:54) = 1) . (77)The alternating Bernoulli function can be represented by the alter-nating zeta function similarly as the Bernoulli function by the zetafunction: B ∗ ( s ) = − sζ ∗ (1 − s ) , ( s (cid:54) = 0) . (78) The alternating Bernoulli polynomials
For a positive real number x define the alternating Hurwitz zetafunction as ζ ∗ ( s, x ) = ∞ (cid:88) n =0 ( − n ( n + x ) s , for Re( s ) > , (79)and for other values of s by analytic continuation. It is connectedwith the Hurwitz Zeta function by ζ ∗ ( s, x ) = 2 − s (cid:18) ζ (cid:16) s, x (cid:17) − ζ (cid:18) s, x + 12 (cid:19)(cid:19) . (80)The alternating Bernoulli rational polynomials are defined as B ∗ n ( x ) = − nζ ∗ (1 − n, x ) . (81)This definition is in analogy to our introduction of the Bernoullipolynomials (27). The alternating Bernoulli numbers are by (42)half the Genocchi numbers: B ∗ n = 2 n − (cid:0) B n (cid:0) (cid:1) − B n (1) (cid:1) =G n / . The connection with the Euler function
The Euler polynomials are closely related to the Bernoullipolynomials. In handbooks of mathematical functions they areoften treated side by side and shown how one can be expressed bythe other. 24ur perspective is a little different. We recall that the interpo-lation of rational sequences is our central theme. However, the twocases already differ concerning whether the numbers are values ofthe respective polynomials: In the Bernoulli case they are, in theEuler case they are not.The generalized
Euler function is defined as a shifted versionof the generalized Genocchi function. G( s, v ) = 2 s (cid:18) B (cid:16) s, v (cid:17) − B (cid:18) s, v + 12 (cid:19)(cid:19) , (82) E( s, v ) = − G( s + 1 , v ) s + 1 . (83)The Euler polynomials are E n ( x ) = E( n, x ) for integers n ≥ . Theone-parameter Euler function is the special case v = 1 . E( s ) = E( s, . (84)The rational numbers E( n ) for integers n ≥ are , , , − , , , , − , , , , − , , , . . . . However, these numbers do not , differently from the case of theBernoulli numbers, inherit the name from their defining function.The
Euler numbers are the values at the integers of the generalizedEuler function (83) at v = , scaled with a power of 2. E c ( s ) = 2 s E (cid:18) s, (cid:19) . (85)These are the equivalents to the central Bernoulli numbers (37).Let us gather the two most important special cases of the Eulerfunction. For the Euler polynomials we have E n ( x ) = 2 ζ ∗ ( − n, x ) = − ∗ n +1 ( x ) n + 1 = − G n +1 ( x ) n + 1 . (86)As a special case we have ∗ n = − n E n − (1) for n ≥ .For the Euler numbers we have E n = 2 n E n (cid:18) (cid:19) = − n +1 B ∗ n +1 (cid:0) (cid:1) n + 1 = − n G n +1 ( ) n + 1 . (87)25 he extended Bernoulli numbers The family tree of the Euler numbers is subdivided into threebranches: the
Euler secant numbers , the
Euler tangent numbers andthe
André numbers . The André numbers are the secant numbersinterwoven with the tangent numbers, that is the term-wise sum ofthe secant and the tangent sequence.The traditional way of naming reserves the name Euler numbersfor the Euler secant numbers, while the way preferred by combinato-rialists [59] calls the unsigned André numbers Euler numbers. Theintroduction of the name ‘André numbers’ in honor of Désiré An-dré who studied their combinatorial interpretation as -alternatingpermutations in 1879 [2] and 1881 [3] resolves this ambiguity. n E tan − −
272 0E sec − −
61 0 1385A 1 1 − − − −
272 1385
Euler and André numbers
Let us try to apply the above extension procedure for the Eulernumbers to the Bernoulli numbers somehow. First, we name theBernoulli numbers with even index the
Bernoulli tangent numbers and those with odd index (provisionally) the lost Bernoulli numbers . n B ? ? / ? − / ? / ? − / The lost Bernoulli numbers
The next table shows what will be the outcome of our choicethat we will define in the next section. n B tan − − B sec − −
0B 1
12 16 − −
130 25992 142 − − The extended Bernoulli numbers n e v e n ][ n o dd ] [ n o dd ][ n e v e n ] ( n − n ) / n ( n − n ) / n n ! n ! / ( n − n ) , , , , , , ,... A nd r ´ e nu m b e r s , , , , , , , ,... E u l e r s e c a n t , , , , , , , ,... E u l e r t a n g e n t , , , , , , , ,... B e r n o u lli s e c a n t , , , , , , , ,... B e r n o u lli t a n g e n t , , , , , , ,... B e r n o u lli e x t e nd e d , , , , , , ,... E u l e r z e t a nu m b e r s T h e E u l e r – B e r n o u ll if a m i l y o f n u m be r s( u n s i g n e d ve r s i o n ) he extended Bernoulli function The extended Bernoulli function is defined as B ( s ) = − s (cid:101) ζ (1 − s ) (88)for s (cid:54) = 0 and the limiting value B (0) = 1 + π/ log(4) , where (cid:101) ζ ( s ) = ζ ( s ) + ζ (cid:0) s, (cid:1) − ζ (cid:0) s, (cid:1) s − . (89)An alternate form to write (88) is B ( s ) = B( s ) + s s − s − (cid:18) ζ (cid:18) − s, (cid:19) − ζ (cid:18) − s, (cid:19)(cid:19) . (90)The extended Bernoulli numbers are the values of the extendedBernoulli function at positive integers for n ≥ and by convention B n = B( n ) for n ∈ { , } (see the table above).The question remains: why did we choose the extended Bernoullinumbers in this manner? The answer is: because of their relationto the Euler zeta numbers, which we will introduce next. Thefigure above indicates the relations between these six sequences byshowing how they derive from a seventh sequence, the Euler zetanumbers. The extended Euler function
The Bernoulli numbers and the Euler numbers have a commonbackbone: the
Euler zeta numbers defined as the values at thepositive integers of the function E ( s ) = 4 s +1 − s +1 ( s + 1)! B ( s + 1) . (91)These numbers, in their unsigned form, were introduced byLeonhard Euler in 1735 in De summis serierum reciprocarum [19].For n ≥ they are , , − , − , , , − , − , , , . . . . . Euler, De summis serierum reciprocarum, 1735. The
André function A ( s ) interpolating the André numbers is A ( s ) = s ! E ( s ) (92) = (4 s +1 − s +1 ) B ( s + 1) s + 1 (93) = (2 s +1 − s +1 ) (cid:101) ζ ( − s ) . (94)When s is a positive integer, these identities are well known, butless known in the generality given here, as identities of complexfunctions. The unsigned extended functions
The unsigned André function has the representation A (cid:63) ( s ) = i ( i s Li − s ( − i ) − ( − i ) s Li − s ( i )) (95)where i is the imaginary unit, Li s ( v ) is the polylogarithm and theprincipal branch of the logarithm is used for the powers.The unsigned André numbers are the values at the nonnegativeintegers, A (cid:63)n = A (cid:63) ( n ) . Its counterpart, the unsigned extendedBernoulli function , is defined as B (cid:63) ( s ) = s A (cid:63) ( s − s − s ( s (cid:54) = 0) . (96)For s = 0 the function B (cid:63) is supplemented by the limiting value,which, surprisingly, is − . Therefore B (cid:63) interpolates the unsignedextended Bernoulli numbers only for n ≥ .29 he unsigned André function A (cid:63) The Swiss knife polynomials
The reader, exhausted from studying all these higher transcen-dental functions, may wonder whether the many numbers consideredhere are also easier to calculate. Fortunately, the answer is ‘yes’,and amazingly a single method is sufficient. κ n ( x ) = n (cid:88) k =0 α ( k + 1)2 (cid:98) k/ (cid:99) k (cid:88) v =0 ( − v (cid:18) kv (cid:19) ( x + v + 1) n . (97)Here α is the repeating sequence period(0, 1, 1, 1, 0, -1, -1, -1).The author dubbed κ n ( x ) the Swiss knife polynomials because theyallow calculating the Euler–Bernoulli family of numbers efficiently.They were introduced in
A153641 and discussed in [47].The coefficients of the polynomials are integers, in contrastto the coefficients of the Euler and Bernoulli polynomials, whichare rational numbers. The Euler, Bernoulli, Genocchi, Euler zeta,tangent as well as the André numbers and the Springer numbersare either values or scaled values of these polynomials, see the tablebelow.The polynomials display a beautiful sinusoidal behavior if suit-able scaled, which can be explained with the Fourier analysis ofthe generalized zeta function. 30 xx − x − xx − x + 5 x − x + 25 xx − x + 75 x − x − x + 175 x − xx − x + 350 x − x + 1385 x − x + 630 x − x + 12465 x The Swiss knife polynomials κ n ( x ) /n ! Euler zeta even κ n (0) / n ! Euler zeta odd κ n (1) / n ! Euler zeta ( − (cid:98) n/ (cid:99) κ n ( n mod 2) / n ! Euler secant κ n (0) Euler tangent κ n (1) Euler extended ( − (cid:98) n/ (cid:99) κ n ( n mod 2) Bernoulli tangent κ n − (1) n / (4 n − n ) Bernoulli secant κ n − (0) n / (4 n − n ) Bernoulli extended κ n − (( n −
1) mod 2) n / (4 n − n ) Genocchi κ n ( − n + 1) / n Springer κ n (1 /
2) 2 n Some applications of the Swiss knife polynomials symptotics for the Bernoulli function An asymptotic expansion of the Bernoulli function results directlyfrom B( s ) = − ζ ( s ) τ ( s ) s ! by using Stirling’s formula and thegeneralized harmonic numbers.For an even positive integer n the Bernoulli function has anefficient asymptotic approximation [44], | B( n ) | ∼ π (cid:16) n π e (cid:17) n +1 / exp (cid:18)
12 + n − − n −
360 + n − (cid:19) . (98)Here we use the coefficients in Stirling’s expansion for log(Γ( s )) ,see A046969 . The number of exact decimal digits guaranteed bythis formula is apparently n ) if n ≥ . This approximationis used in the Boost C++ library [7] for very large arguments n .For general s > the asymptotic expansion is B( s ) ∼ − π cos (cid:16) sπ (cid:17) (cid:16) s π e (cid:17) s +1 / R( s ) , R( s ) = exp (cid:32)
12 + ∞ (cid:88) n =1 B( n + 1) n + 1 s − n n (cid:33) . (99)If one agrees to the convention to read s − = 1 , then one can startthe sum at n = 0 and do without the constant term .Other asymptotic developments can be based on different devel-opments of the Gamma function, for instance on Binet’s formula[18, p. 48] generalized by Gergő Nemes [52, 4.2]. More generalasymptotic expansions and error bounds follow from those of theHurwitz zeta function that Nemes established in [53]. Epilogue: generating functions
The value of
B(1) deserves special attention. Since it is wellknown that (cid:80) ∞ j =0 γ j /j ! = 1 / it follows from (8) that B(1) = 1 / .Unfortunately the popular generating function z/ (e z − missesthis value and disrupts at this point the connection between theBernoulli numbers and the ζ function.32or those who do not care about the connection with the zetafunction, we add: Even the most elementary relations between theBernoulli numbers and the Bernoulli polynomials break with thischoice. For instance, consider the basic identity (39). It applies toall Bernoulli numbers if B n = B n (1) but not if B n = B n (0) is set.Instead use the power series f ( z ) with constant term 1 such thatthe coefficient of x n in ( f ( x )) n +1 equals 1 for all n . There is onlyone power series satisfying this condition, as Friedrich Hirzebruch[29] observes. This series is the Todd function (called after JohnArthur Todd) T( z ) = z − e − z = 1 + 12 z
1! + 16 z − z
4! + . . . (100)and generates the Bernoulli numbers matching the values of theBernoulli function at the nonnegative integers.A modern exposition based on the Todd series is the monograph[5]. The authors adopt this definition “because it is the originaldefinition of Seki and Bernoulli for one thing, and it is better suitedto the special values of the Riemann zeta function for another.”Similarly, J. Neukirch in
Algebraic Number Theory [54] writes: Thedefinition f ( z ) = z/ (1 − e − z ) “is more natural and better suited forthe further development of the theory.” One might hope that allmathematicians will use this consistent definition one day. Fair use of ‘Don Quixote’, sketch by Pablo Picasso.
Acknowledgments
The author thanks Jörg Arndt, Petros Hadjicostas, VáclavKotěšovec, Gergő Nemes and Michael Somos for their reading andproviding valuable feedback on an earlier version of this manuscript.Without using Neil Sloane’s
OEIS this essay could have beenwritten, but it would only have been half as much fun.33 ey words:
Bernoulli function, Bernoulli constants, Bernoulli num-bers, Bernoulli cumulants, Bernoulli functional equation, Bernoullicentral function, extended Bernoulli function, Bernoulli centralpolynomials, alternating Bernoulli function, Stieltjes constants,Riemann zeta function, Hurwitz zeta function, Worpitzky num-bers, Worpitzky transform, Hasse representation, Hadamar product,Genocchi numbers, André numbers, Swiss knife polynomials.2020 MSC. Primary 11B68, Secondary 11M35.OEIS:
A164555 / A027642 , A019692 , A028246 , A163626 , A226158 , A263634 , A301813 , A303638 , A318259 , A333303 , A335263 , A335750 , A335751 , A335947 , A335948 , A335949 , A336454 , A336517 , A335953 .Code repository and supplements, GitHub [46].
ORCID 0000-0001-6245-708X
List of Figures γ/π . . . . . . . . . . . . 84 The Hurwitz–Bernoulli function . . . . . . . . . . . 125 The central Bernoulli function . . . . . . . . . . . . 136 Derivatives of the Bernoulli function . . . . . . . . 157 Euler’s γ and B (cid:48) ( s ) . . . . . . . . . . . . . . . . . . 168 The Hadamar decomposition . . . . . . . . . . . . 229 The Euler–Bernoulli number family . . . . . . . . . 2710 L. Euler: De summis serierum . . . . . . . . . . . . 2911 The unsigned André function . . . . . . . . . . . . 3012 The Swiss knife polynomials . . . . . . . . . . . . . 3113 Pablo Picasso, Don Quixote . . . . . . . . . . . . . 3314 The Riemann zeros . . . . . . . . . . . . . . . . . . 3615 Phase portrait of B( x ) . . . . . . . . . . . . . . . . 3616 Complex view of B( x ) . . . . . . . . . . . . . . . . 3717 3-dim view of B( x ) . . . . . . . . . . . . . . . . . . 3718 Logarithm of B( x ) . . . . . . . . . . . . . . . . . . 3822 The Riemann ξ -factor of B( x ) . . . . . . . . . . . . 3923 The singularity factor of B( x ) . . . . . . . . . . . . 3924 Jensen’s table from 1895 . . . . . . . . . . . . . . . 4634 ndex of Definitions alternating Bernoullipolynomials, 24function, 23numbers, 23alternating zetaHurwitz function, 24Riemann function, 24Andréfunction, 29numbers, 29unsigned function, 29Bernoulliasymptotics, 32central function, 13central numbers, 13central polynomials, 14constants, 5cumulants, 18derivative, 15extended function, 28function, 5functional equation, 21generalized constants, 10generalized function, 10Hurwitz function, 11log. derivative, 17lost numbers, 26numbers, 5polynomials, 10unsigned function, 29Constant γ/π , 9 γ Euler, 4, 17 τ , 20Dirichlet eta function, 24Eulerextended function, 28function, 25 numbers, 25polynomials, 25zeta numbers, 28generating functioninconsistent, 32Todd, 33Genocchifunction, 14numbers, 15polynomials, 15Hadamard product, 23Hasse representation, 20Hurwitzformula, 12zeta function, 9Jensenformula, 11, 22logarithmicderivative, 17polynomials, 18Polynomialscentral Bernoulli, 14Genocchi, 15Swiss knife, 30Worpitzky, 19Riemann ξ function, 21 ζ -function, 4extended function, 28hypothesis, 17Stieltjesconstants, 4gen. constants, 9tau function, 20Worpitzkygen. transform, 19numbers, 18transform, 1835 ernoulli function and Riemann zeros on the critical line.Phase portrait of the Bernoulli function on the right half plane. he Bernoulli function on the right half plane, complex view.The Bernoulli function on the right half plane, 3-dim view. he logarithm of the Bernoulli function on the right half plane. The red peaks on the x-axis correspond to the real zero of theBernoulli function (the vanishing of the odd Bernoulli numbers).The front side of the plot shows the logarithm of the Bernoullifunction on the critical line.The Hadamard decomposition of the Bernoulli function is displayedin the two plots below. 38 he Hadamard decomposition of log B : the Riemann ξ -factor.The Hadamard decomposition of log B : the singularity factor. eferences [1] T. M. Apostol , Zeta and related functions, chapter 25 of the DigitalLibrary of Mathematical Functions (DLMF), release 1.0.18 of 2018-03-27, https://dlmf.nist.gov/25.11 .[2]
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J. Worpitzky , Studien über die Bernoullischen und EulerschenZahlen, Journal für die reine und angewandte Mathematik (94), pp.203–232, (1883). ernoulli constants β n b n = β n n ! [ 0] +1.00000000000000000000000e+00 +1.00000000000000000000000e+00[ 1] -5.77215664901532860606512e-01 -5.77215664901532860606512e-01[ 2] +1.45631690967353449721173e-01 +7.28158454836767248605864e-02[ 3] +2.90710895786169554535912e-02 +4.84518159643615924226519e-03[ 4] -8.21533768121338346464019e-03 -3.42305736717224311026674e-04[ 5] -1.16268503273365002873409e-02 -9.68904193944708357278404e-05[ 6] -4.75994290380637621052001e-03 -6.61103181084218918127779e-06[ 7] +1.67138541801139726910695e-03 +3.31624090875277235933919e-07[ 8] +4.21831653646200836859278e-03 +1.04620945844791874221051e-07[ 9] +3.16911018422735558641847e-03 +8.73321810027379736116201e-09[10] +3.43947744180880481779146e-04 +9.47827778276235895555407e-11[11] -2.25866096399971274152095e-03 -5.65842192760870796637242e-11[12] -3.24221327452684232007482e-03 -6.76868986351369665586675e-12[13] -2.17454785736682251359552e-03 -3.49211593667203185445522e-13[14] +3.84493292452642224040106e-04 +4.41042474175775338023724e-15[15] +3.13813893088949918755710e-03 +2.39978622177099917550506e-15[16] +4.53549848512386314628695e-03 +2.16773122007268285496389e-16[17] +3.39484659125248617003234e-03 +9.54446607636696517342499e-18[18] -4.72986667978530060590399e-04 -7.38767666053863649781558e-20[19] -5.83999975483580370526234e-03 -4.80085078248806522761766e-20[20] -1.00721090609471125811119e-02 -4.13995673771330564126948e-21[21] -9.79321479174274843741249e-03 -1.91682015939912339496482e-22[22] -2.29763093463200254783750e-03 -2.04415431222621660772759e-24[23] +1.24567903906919471380695e-02 +4.81849850110735344392922e-25[24] +2.98550901697978987031938e-02 +4.81185705151256647946111e-26[25] +3.97127819725890390476549e-02 +2.56026331031881493660913e-27[26] +2.79393907712007094428316e-02 +6.92784089530466712388013e-29[27] -1.77336950032031696506289e-02 -1.62860755048558674407104e-30[28] -9.73794335813190698522061e-02 -3.19393756115325557604211e-31[29] -1.85601987419318254285110e-01 -2.09915158936342552768549e-32[30] -2.21134553114167174032372e-01 -8.33674529544144047562508e-34[31] -1.10289594522767989385320e-01 -1.34125937721921866750473e-35[32] +2.40426431930087325860325e-01 +9.13714389129817199794565e-37[33] +8.48223060577873259185100e-01 +9.76842144689316562821221e-38[34] +1.53362895967472747676942e+00 +5.19464288745573322360277e-39[35] +1.78944248075279625487765e+00 +1.73174959516100441594633e-40[36] +7.33429569739007257407068e-01 +1.97162023326628724184554e-42[37] -2.68183987622201934815062e+00 -1.94848008275558832944550e-43[38] -8.96900252642345710339731e+00 -1.71484028164349789185891e-44[39] -1.67295744090075569567376e+01 -8.20162325795024844398325e-46[40] -2.07168737077169487591557e+01 -2.53909617003982347034561e-47[41] -1.01975839351792340684642e+01 -3.04837480681247325379474e-49[42] +3.02221435698546147289327e+01 +2.15103288078139524274228e-50[43] +1.13468181875436585038896e+02 +1.87813767783170614584043e-51[44] +2.31656933743621048467685e+02 +8.71456987091534575015368e-53[45] +3.23493565027658727705394e+02 +2.70429325494165277375631e-54[46] +2.33327851136153134645751e+02 +4.24030297482975157602467e-56[47] -3.10666033627557393250479e+02 -1.20123014394335413203436e-57[48] -1.63390919914361991588774e+03 -1.31619164554817482023699e-58[49] -3.85544150839888666284589e+03 -6.33824832920169119558284e-60[50] -6.29221938159892345466820e+03 -2.06884990450560309799972e-61 . Jensen computing the Bernoulli function B( − z ) in 1895. Bernoulli cumulants − C n − C n n ! [ 0] 5.77215664901532860606512e-01 5.77215664901532860606512e-01[ 1] 1.87546232840365224597203e-01 1.87546232840365224597203e-01[ 2] 1.03377264066385787604016e-01 5.16886320331928938020082e-02[ 3] 8.85099529527224643874814e-02 1.47516588254537440645802e-02[ 4] 1.08587469323889089789907e-01 4.52447788849537874124612e-03[ 5] 1.73615424543021976826038e-01 1.44679520452518314021698e-03[ 6] 3.39511736293491636244455e-01 4.71544078185405050339520e-04[ 7] 7.82108682587720478889760e-01 1.55180294164230253747968e-04[ 8] 2.07023895260357186095142e+00 5.13452121181441433767714e-05[ 9] 6.18396764525550941770221e+00 1.70413570471106410320277e-05[10] 2.05609655822716770310295e+01 5.66605092104047537230752e-06[11] 7.52770418698025254867191e+01 1.88584861185772720976429e-06[12] 3.00839552402986587830834e+02 6.28055422785616139551170e-07[13] 1.30294506447394086233815e+03 2.09240519073573812751380e-07[14] 6.07848047275118529922938e+03 6.97247031236967546712980e-08[15] 3.03866350907681039525268e+04 2.32371573798165201572008e-08[16] 1.62043648652105132185893e+05 7.74483945590082160394406e-09[17] 9.18184885317620501162217e+05 2.58143755665663419434431e-09[18] 5.50888477406867075105888e+06 8.60444114522719419372965e-10[19] 3.48887511550119854518536e+07 2.86807697455596821291504e-10[20] 2.32588267070109667252423e+08 9.56011653113908225323882e-11[21] 1.62810356589926860813867e+09 3.18667751404435348670453e-11[22] 1.19393632362359042013562e+10 1.06222024071484413115002e-11[23] 9.15348288139870681527169e+10 3.54072294392050811222228e-12[24] 7.32277241973702388310021e+11 1.18023874334765961275615e-12[25] 6.10230340797500863040914e+12 3.93412466914448230705943e-13[26] 5.28865934416774000146351e+13 1.31137399472672361405551e-13[27] 4.75979146079582837193300e+14 4.37124485922912446959546e-14[28] 4.44247093871183417895415e+15 1.45708126178764844543513e-14[29] 4.29438794111778364421710e+16 4.85693682340507565625935e-15[30] 4.29438756134456760579530e+17 1.61897879796099259670507e-15 nsigned extended Bernoulli and Euler zeta numbers n B (cid:63)n E (cid:63)n
16 12
356 13
130 524
142 61720
130 2778064
566 505213628800
76 19936098187178291200 rologue: extension by interpolation 1Synopsis 3The Stieltjes constants 4The Bernoulli constants 5The Bernoulli numbers 5The expansion of the Bernoulli function 6Integral formulas for the Bernoulli constants 7Some special integral formulas 8Generalized Stieltjes and Bernoulli constants 9The generalized Bernoulli function 10Integral formulas for the Bernoulli function 10The Hurwitz–Bernoulli function 11The central Bernoulli function 12The central Bernoulli polynomials 13The Genocchi function 14Derivatives of the Bernoulli function 15The logarithmic derivative 17Bernoulli cumulants 18Worpitzky numbers 18The generalized Worpitzky transform 19The Hasse representation 20The constant tau 20The functional equation 21Representation by the Riemann ξ function 21The Hadamard decomposition 22The alternating Bernoulli function 23The alternating Bernoulli polynomials 24The connection with the Euler function 24The André numbers 26The extended Bernoulli function 28The extended Euler function 28The unsigned extended functions 29The Swiss knife polynomials 30Asymptotics for the Bernoulli function 32Epilogue: generating functions 32Index of definitions 35Plots 40References 40Appendix: Bernoulli constants & cumulants 45function 21The Hadamard decomposition 22The alternating Bernoulli function 23The alternating Bernoulli polynomials 24The connection with the Euler function 24The André numbers 26The extended Bernoulli function 28The extended Euler function 28The unsigned extended functions 29The Swiss knife polynomials 30Asymptotics for the Bernoulli function 32Epilogue: generating functions 32Index of definitions 35Plots 40References 40Appendix: Bernoulli constants & cumulants 45