A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer m
aa r X i v : . [ m a t h . C A ] J a n A Poisson-Jacobi-type transformation for the sum P ∞ n =1 n − m exp( − an ) for positive integer m R. B. Paris
School of Engineering, Computing and Applied Mathematics , University of Abertay Dundee, Dundee DD1 1HG, UK
Abstract
We obtain an asymptotic expansion for the sum S ( a ; w ) = ∞ X n =1 e − an n w as a → | arg a | < π for arbitrary finite w >
0. The result when w = 2 m , where m isa positive integer, is the analogue of the well-known Poisson-Jacobi transformation for thesum with m = 0. Numerical results are given to illustrate the accuracy of the expansion. Mathematics Subject Classification:
Keywords:
Poisson-Jacobi transformation, asymptotic expansion, inverse factorial ex-pansion
1. Introduction
The classical Poisson-Jacobi transformation is given by ∞ X n =1 e − an = 12 r πa −
12 + r πa ∞ X n =1 e − π n /a , (1.1)where the parameter a satisfies ℜ ( a ) >
0. This transformation relates a sum of Gaussianexponentials involving the parameter a to a similar sum with parameter π /a . In the case a → ℜ ( a ) >
0, the convergence of the sum on the left-hand side becomes slow, whereas thesum on the right-hand side converges rapidly in this limit. Various proofs of the well-knownresult (1.1) exist in the literature; see, for example, [3, p. 120], [4, p. 60] and [5, p. 124].In this note we consider the sum S ( a ; w ) = ∞ X n =1 e − an n w ( ℜ ( a ) > . (1.2)This sum converges for any finite value of the parameter w provided ℜ ( a ) >
0; when a = 0then S (0; w ) reduces to the Riemann zeta function ζ ( w ) when ℜ ( w ) >
1. Consequently, theseries in (1.2) can be viewed as a smoothed Dirichlet series for ζ ( w ). The asymptotic expansion1 R. B. Paris of S ( a ; w ) as a → ℜ ( a ) > w = 2 m , where m is a positive integer, for which we establish a transformation for S ( a ; 2 m )analogous to that in (1.1) valid as a → ℜ ( a ) >
0. This similarly involves the series in(1.2) with a replaced by π /a , but with each term decorated by an asymptotic series in a . Arecent application of the series with w = 2 and w = 4 has arisen in the geological problem ofthermochronometry in spherical geometry [6].
2. An expansion for S ( a ; w ) as a → when w = 2 , , . . . Our starting point is the well-known Cahen-Mellin integral (see, for example, [3, § z − α e − z = 12 πi Z c + ∞ ic −∞ i Γ( s − α ) z − s ds ( z = 0 , | arg z | < π ) , (2.1)where c > ℜ ( α ) so that the integration path passes to the right of all the poles of Γ( s − α )situated at s + α − k ( k = 0 , , , . . . ). For simplicity in presentation we shall assume throughoutreal values of w >
0. Then, it follows that S ( a ; w ) = ∞ X n =1 e − an n w = ∞ X n =1 n − w πi Z c + ∞ ic −∞ i Γ( s )( an ) − s ds = 12 πi Z c + ∞ ic −∞ i Γ( s ) ζ (2 s + w ) a − s ds, upon reversal of the order of summation and integration, which is justified when c > max { , − w } , and evaluation of the inner sum in terms of the Riemann zeta function. The integrandpossesses simple poles at s = − w and s = − k ( k = 0 , , , . . . ), except if w = 2 m + 1 is anodd positive integer when the pole at s = − w is double. The case when w = 2 m is an evenpositive integer requires a separate investigation which is discussed in Section 3.Consider the integral taken round the rectangular contour with vertices at c ± iT , − c ′ ± iT ,where c ′ >
0. The contribution from the upper and lower sides s = σ ± iT , − c ′ ≤ σ ≤ c ,vanishes as T → ∞ provided | arg a | < π , since from the behaviourΓ( σ ± it ) = O ( t σ − e − πt ) , ζ ( σ ± it ) = O ( t µ ( σ ) log A t ) , ( t → ∞ ) , where for σ and t real µ ( σ ) = 0 ( σ > , − σ (0 ≤ σ ≤ , − σ ( σ < ,A = 1 (0 ≤ σ ≤ , A = 0 otherwise , the modulus of the integrand is controlled by O ( T σ + µ ( σ ) − log T e − ∆ T ), with ∆ = π − | arg a | .The residue at the double pole s = − m when w = 2 m + 1 ( m = 0 , , , . . . ) is given by( − a ) m m ! { γ − log a + ψ ( m + 1) } , where γ is Euler’s constant and ψ ( x ) is the logarithmic derivative of the gamma function.Displacement of the integration path to the left over the poles then yields (provided w = 2 m ) S ( a ; w ) = J ( a ; w ) + N − X k =0 ′ ( − ) k k ! ζ ( w − k ) a k + R N , (2.2) oisson-Jacobi-type transformation J ( a ; w ) = Γ( − w ) a ( w − / ( w = 2 m + 1)( − a ) m m ! { γ − log a + ψ ( m + 1) } ( w = 2 m + 1) , ,N is a positive integer such that N > w + and the prime on the sum over k denotes theomission of the term corresponding to k = m when w = 2 m + 1.The remainder R N is R N = 12 πi Z − c + ∞ i − c −∞ i Γ( s ) ζ ( w + 2 s ) a − s ds, c = N − . (2.3)It is shown in the appendix, when w = 2 , , . . . , that R N = O ( a N − ) as a → | arg a | < π ,with the constant implied in the O -symbol growing at least like Γ( N +1 − w ). This establishesthat the above series over k diverges as N → ∞ and that (2.2) is therefore an asymptoticexpansion.We remark that the algebraic expansion (2.2) also contains a subdominant exponentiallysmall component as a →
0; compare [3, § w = 0. We do notconsider this further in the present paper.
3. An expansion for S ( a ; 2 m ) when m = 1 , , . . . The case w = 2 m , where m is a positive integer, is more interesting as this leads to theanalogue of the Poisson-Jacobi transformation (1.1). There is now only a finite set of poles ofthe integrand in (2.1) at s = − w and s = 0 , − , − , . . . , − m , since the poles of Γ( s ) at s = − m − k ( k = 1 , , . . . ) are cancelled by the trivial zeros of the zeta function ζ (2 m + 2 s )at s = − m − , − m − , . . . . This has the consequence that the integrand is holomorphic in ℜ ( s ) < − m , so that further displacement of the contour can produce no additional algebraicterms in the expansion of S ( a ; 2 m ). Thus, we find when w = 2 mS ( a ; 2 m ) = Γ( − m ) a m − + m X k =0 ( − ) k k ! ζ (2 m − k ) a k + I L , (3.1)where, upon making the change of variable s → − s , I L = 12 πi Z L Γ( − s ) ζ (2 m − s ) a s ds (3.2)and L denotes a path parallel to the imaginary axis with ℜ ( s ) > m .We now employ the functional relation for ζ ( s ) given by [5, p. 269] ζ ( s ) = 2 s π s − ζ (1 − s )Γ(1 − s ) sin πs (3.3)to convert the argument of the zeta function in (3.2) into one with real part greater than unity.The integral in (3.2) can then be written in the form( − ) m (2 π ) m πi Z L ζ (2 s − m + 1) Γ(2 s − m + 1)Γ( s + 1) (cid:16) a π (cid:17) s ds. Since on the integration path ℜ (2 s − m + 1) >
1, we can expand the zeta function and reversethe order of summation and integration to obtain I L = ( − ) m π m − ∞ X n =1 n m − K n ( a ; m ) , (3.4) R. B. Paris where K n ( a ; m ) := 12 πi Z L Γ( s − m + )Γ( s − m + 1)Γ( s + 1) (cid:16) aπ n (cid:17) s ds, and we have employed the duplication formula for the gamma functionΓ(2 z ) = 2 z − π − Γ( z )Γ( z + ) . The integrals K n ( a ; m ) have no poles in the half-plane ℜ ( s ) > m , so that we can displacethe path L as far to the right as we please. On such a displaced path | s | is everywhere large.The quotient of gamma functions may then be expanded by making use of the result given in[3, p. 53] Γ( s − m + )Γ( s − m + 1)Γ( s + 1) = M − X j =0 ( − ) j c j Γ( s + ϑ − j ) + ρ M ( s )Γ( s + ϑ − M ) (3.5)for positive integer M , where ϑ = − m , c j = ( m ) j ( m + ) j j ! = 2 − j (2 m ) j j !and ρ M ( s ) = O (1) as | s | → ∞ in | arg s | < π . Substitution of this expansion into the integrals K n ( a ; m ) then produces K n ( a ; m ) = M − X j =0 ( − ) j c j πi Z L Γ( s + ϑ − j ) (cid:16) aπ n (cid:17) s ds + R M = M − X j =0 ( − ) j c j (cid:16) aπ n (cid:17) m + j − e − π n /a + R M (3.6)by (2.1), where R M = 12 πi Z L ρ M ( s )Γ( s + ϑ − M ) (cid:16) aπ n (cid:17) s ds. Bounds for the remainder R M have been considered in [3, p. 71, Lemma 2.7], where it is shownthat R M = O (cid:18)(cid:16) aπ n (cid:17) M − ϑ e − π n /a (cid:19) (3.7)as a → | arg a | < π .Collecting together the results in (3.2), (3.4), (3.6) and (3.7), we obtain I L = ( − ) m (cid:16) aπ (cid:17) m − ∞ X n =1 e − π n /a n m M − X j =0 c j (cid:18) − aπ n (cid:19) j + O (cid:18)(cid:16) aπ n (cid:17) M (cid:19) . From (3.1) we now have the following theorem:
Theorem 1 . Let m and M be positive integers. Then, when w = 2 m , we have the expansionvalid as a → in | arg a | < πS ( a ; 2 m ) = Γ( − m ) a m − + m X k =0 ( − ) k k ! ζ (2 m − k ) a k oisson-Jacobi-type transformation − ) m (cid:16) aπ (cid:17) m − ∞ X n =1 Υ n ( a ; m ) n m e − π n /a , (3.8) where Υ n ( a ; m ) has the asymptotic expansion Υ n ( a ; m ) = M − X j =0 ( m ) j ( m + ) j j ! (cid:18) − aπ n (cid:19) j + O (cid:18)(cid:16) aπ n (cid:17) M (cid:19) . This is the analogue of the Poisson-Jacobi transformation in (1.1). In the case m = 0, thequotient of gamma functions in (3.5) is replaced by the single gamma function Γ( s + ), withthe result that c = 1, c j = 0 ( j ≥
1) and Υ n ( a ; m ) = 1 for all n ≥
1. Then (3.8) reduces to(1.1) and is valid for all values of the parameter a (not just a →
0) satisfying | arg a | < π . Remark 1.
We note that the values of the zeta function appearing in (3.8) can be expressedalternatively in terms of Bernoulli numbers by the result [2, p. 605] ζ (2 n ) = (2 π ) n n )! | B n | .
4. Numerical results and concluding remarks
From the well-known values [2, p. 605] ζ (2) = π , ζ (4) = π , we obtain from Theorem 1 the expansions in the cases m = 1 and m = 2 given by S ( a ; 2) = π a − ( πa ) − (cid:16) aπ (cid:17) ∞ X n =1 e − π n /a n (cid:26) M − X j =0 ( ) j (cid:18) − aπ n (cid:19) j + O ( a M ) (cid:27) (4.1)and S ( a ; 4) = π − π a − a π a + (cid:16) aπ (cid:17) ∞ X n =1 e − π n /a n (cid:26) M − X j =0 ( ) j (2) j j ! (cid:18) − aπ n (cid:19) j + O ( a M ) (cid:27) (4.2)valid as a → | arg a | < π .In Table 1 we show the results of numerical calculations for the case m = 2. For differentvalues of the parameter a we present the value of the absolute error in the computation of S ( a ; 4) from (4.2). In the computations, we have used only the n = 1 term (since the order of exp( − π /a ) was found to be less than the error), with the expansion for Υ ( a ; 2) optimallytruncated (corresponding to truncation at, or near, the least term in modulus) at index j ≃ ( π /a ) − . It is seen that the error when a = 0 . a ≃ S p ( a ; w ) ≡ ∞ X n =1 e − an p n w ( a → , ℜ ( a ) > R. B. Paris
Table 1:
Values of the absolute error in the computation of S ( a ; 4) from (4.2). The value of the index j corresponds to optimal truncation of the expansion Υ ( a ; 2). a S ( a ; 4) Error j . × − . × − . × − . × − . × − . × − . × − . × − w and p . The case w = 0 and p >
0, corresponding to the Euler-Jacobiseries, has been considered in [3, § p is arational fraction. The details of the small- a expansion of S p ( a ; w ) will be presented elsewhere. Acknowledgement.
The author wishes to acknowledge B. Guralnik for having brought thisproblem to his attention
Appendix: A bound for the remainder R N Let ψ = arg a and integer N > w + . Upon replacement of s by − s followed by use of (3.3),the remainder R N in (2.3) becomes R N = (2 π ) w πi Z N − + ∞ iN − −∞ i ζ (1 − w + 2 s ) Γ(1 − w + 2 s )Γ(1 + s ) sin π ( s − w )sin πs (cid:16) a π (cid:17) s ds. With s = N − + it , t ∈ ( −∞ , ∞ ) we have | R N | ≤ (2 π ) w − (cid:16) a π (cid:17) N − ζ (2 N − w ) Z ∞−∞ e − ψt (cid:12)(cid:12)(cid:12)(cid:12) Γ(2 N − w + 2 it )Γ( N + + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt, since | ζ ( x + it ) | ≤ ζ ( x ) ( x >
1) and (cid:12)(cid:12)(cid:12)(cid:12) sin π ( N − − w + it )sin π ( N + + it ) (cid:12)(cid:12)(cid:12)(cid:12) = | cos π ( w − it ) | cosh πt = (cos πw + sinh πt ) cosh πt ≤ . It then follows that | R N | = O (cid:18)(cid:16) aπ (cid:17) N − Z ∞−∞ e − ψt | Γ( N − w + it ) | (cid:12)(cid:12)(cid:12)(cid:12) Γ( N − w + + it )Γ( N + + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:19) . (A.1)Using the argument presented in [3, p. 126], we set N − w − = M + δ , with − < δ ≤ so that M ≤ N −
1, to find (cid:12)(cid:12)(cid:12)(cid:12) Γ( N − w + + it )Γ( N + + it ) (cid:12)(cid:12)(cid:12)(cid:12) = P ( t ) g ( t ) , g ( t ) := (cid:12)(cid:12)(cid:12)(cid:12) Γ(1 + δ + it )Γ( + it ) (cid:12)(cid:12)(cid:12)(cid:12) , oisson-Jacobi-type transformation P ( t ) = ( + t ) − Q Mr =1 { ( r + δ ) + t } Q N − r =1 { ( r + ) + t } ≤ ( + t ) − ≤ . From the upper bound for the gamma function Γ( z ) with z = x + it , x > | Γ( z ) | < Γ( x )(1 + t /x ) x − e −| t | φ ( t ) e / (6 | z | ) , φ ( t ) = arctan( | t | /x ) < Γ( x )(1 + τ ) x − exp [ x { ω ( τ ) − π | τ |} e / (6 x ) < e x Γ( x )(1 + τ ) x − e − πx | τ | e / (6 x ) , where we have put τ = t/x , defined ω ( τ ) = | τ | arctan(1 / | τ | ) and used the fact that 0 ≤ ω ( τ ) < τ ∈ [0 , ∞ ), with the limit 1 being approached as τ → ∞ . Substituting the above boundsinto (A.1), we see on setting x = N − w that | R N | = e N Γ( N − w +1) O (cid:18)(cid:16) aπ (cid:17) N − Z ∞ (1 + τ ) N/ g ( τ ) { e − ∆ + τ + e − ∆ − τ } dτ (cid:19) , where ∆ ± = ( N − w )( π ± ψ ). Since g ( τ ) = O ( τ δ + ) as τ → ∞ , the integral is convergentprovided | ψ | < π and is manifestly an increasing function of N .Hence R N = O ( a N − ) ( a → , | arg a | < π ) , (A.2)with the constant implied in the O -symbol growing at least like Γ( N +1 − w ) as N increases. References [1] V. Kowalenko, N. E. Frankel, M. L. Glasser and T. Taucher,
Generalised Euler-Jacobi Inversion Formulaand Asymptotics Beyond All Orders , London Math. Soc. Lecture Notes Series , Cambridge UniversityPress, Cambridge, 1995.[2] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.),
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