2  Partial-fraction decomposition

Lemma 1 If m_j is an integer and |rz_j| < 1 for j = 1,¼,n , then

n Õ j=1  (1-rzj)mj = ¥ å s=0  Ts rs ,

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T0 = 1 ,       Ts = å  H1a1¼Hsas a1 ! ¼as !  ,       Hs =  -1 s   n å j=1   mj zjs ,        s ³ 1,

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where the sum defining T_s extends over all nonnegative integers a₁ ,¼,a_s such that a₁ + 2a₂ + ¼+ sa_s = s. If m_j Î {0,1} for j=1,¼,n then (-1)^s T_s is by definition the elementary symmetric function E_s of degree s in m₁ z₁ ,¼,m_n z_n .

Using the power-series expansion of a logarithm, we see that

ln n Õ j=1  (1-rzj)mj = n å j=1  mj ln(1-rzj) = - n å j=1  mj  ¥ å s=1   rs zjs s = ¥ å s=1  Hs rs .

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To obtain the coefficient T_s of r^s in

n Õ j=1  (1-rzj)mj = exp ¥ å s=1  Hs rs = ¥ Õ k=1  exp(Hk rk) ,

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we omit factors in the infinite product with k > s, multiply the exponential series for exp(H₁ r),¼,exp(H_s r^s), and pick out the coefficient of r^s.

Definition 1 Define

M= n å j=1  mj ,       B = n Õ j=1  bjmj,       dij = ai bj -aj bi ,       Di = n Õ [( j=1) || (j ¹ i)]  dji mj ,

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m±s(i) =  -1 s n å [( j=1) || (j ¹ i)]  mj æ è  dij bj ö ø ±s    ,       s = 1,2,3,¼,

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C0(i)=1,       C±s(i) = å  m±1a1(i)¼ m±sas(i) a1 ! ¼as !  ,       s = 1,2,3,¼,

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where upper (lower) signs go together and the last sum extends over all nonnegative integers a₁,¼,a_s such that a₁ +2a₂ +¼+ sa_s = s. In particular we have

C±1(i) = m±1(i) ,       C±2(i) =  1 2   m2±1(i)+m±2(i) ,       C±3(i) =  1 6  m3±1(i) + m±1(i) m±2(i) + m±3(i) .       (10) C±4(i) =  1 24  m4±1(i) +  1 2  m2±1(i) m±2(i) + m±1(i)m±3(i) +  1 2  m2±2(i) +m±4(i) .

We note that d_ij ¹ 0 if i ¹ j because of the assumption that no two linear factors on the right side of (1.2) are proportional.

Theorem 1 The decomposition of P(t)/Q(t) into partial fractions is

P(t) Q(t) = n Õ j=1  (aj + bj t)mj = B bi-M  M å q=0  CM-q(i) (ai +bi t)q + n å i=1  Di bimi -M -mi å q=1   Cmi +q(i) (ai+bi t)-q .    (11)

The polynomial part is independent of the choice of i, and each sum over q is empty if the upper limit is less than the lower limit.

The definition (2.5) of d_ij implies

aj +bj t =  bj bi  (ai + bi t) æ è 1 -  dij bj(ai +bi t) ö ø  ,

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whence

n Õ j=1  (aj + bj t)mj = B bi-M (ai + bi t)M  n Õ j=1    æ è 1 -  dij bj(ai +bi t) ö ø mj   .

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This equation is independent of the choice of i. If |t| is sufficiently large, we can expand the last product by Lemma 2.1 with r = 1/(a_i +b_i t) and z_j = d_ij/b_j . It follows that H_s is m_+s(i) as defined by (2.6), the term in H_s with j=i being irrelevant because d_ii=0. Thus we see that T_s is C_+s(i) as defined by (2.7), and (2.11) becomes

P(t) Q(t) = B bi-M ¥ å s=0  Cs(i)(ai +bit)M-s.

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Since this is valid for sufficiently large t, the polynomial part of P/Q consists of the terms with 0 £ s £ M, which are absent unless M ³ 0. By putting s=M-q these become the first term on the right side of (2.9).

To find the rest of (2.9), which contains poles arising from negative m's, we replace (2.10) by

aj + bj t =  dji bi æ è 1-  bj dij   (ai +bi t) ö ø  ,    j ¹ i ,

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whence

n Õ j=1  (aj +bj t)mj = Di bimi -M (ai +bi t)mi  n Õ [( j=1) || (j ¹ i)]    æ è 1-  bj dij  (ai +bi t) ö ø mj   .

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If |a_i +b_i t| is sufficiently small, we can expand the last product by Lemma 2.1 with r = a_i +b_i t and z_j = b_j/d_ij. It follows that H_s is m_-s(i) as defined by (2.6) and that T_s is C_-s(i) as defined by (2.7). Hence (2.11) becomes

P(t) Q(t) = Di bimi -M  ¥ å s=0  C-s(i) (ai +bi t)mi + s .

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Since this is valid if |a_i +b_i t| is sufficiently small, the part of P/Q representing a pole at -a_i/b_i consists of the terms with 0 £ s £ -m_i -1, which are absent unless -m_i ³ 1. These can be rewritten by putting s = -m_i -q as

Di bimi -M  -mi å q=1  Cmi +q(i) (ai +bi t)-q.

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Summation over i takes account of all the poles of P/Q and yields the second term on the right side of (2.9).

We shall now apply the partial-fraction decomposition (2.9) to the integral (1.2), designated henceforth by

I(m) = ó õ x y  h Õ i=1  (ai +bi t)-1/2 n Õ j=1  (aj +bj t)mj dt ,

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where m=(m₁,¼,m_n) is an n-tuple of integers. We define n-tuples

e0 = (0,0,¼,0), e1 = (1,0,¼,0), (20) : en = (0,¼,0,1).

Substitution of (2.9) in (2.17) reduces I(m) to a set of simpler integrals in which at most one component of m is nonzero:

I(m) = B bi-M  M å q=0  CM-q(i) I(qei) + n å i=1   Di  bimi -M  -mi å q=1   Cmi +q(i) I(-qei) .

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The first term on the right side is independent of the choice of i, and each sum over q is empty if the upper limit is less than the lower limit.