In the 1980s the architect Michael Graves proposed demolishing the flanking brownstones down to the 74th Street corner for a complementary addition.
Substituting 1 − y for y and μ − 1 for λμ in (3.26) and using the complementary addition theorem for Apostol-Bernoulli polynomials, we get our result.
Therefore, to explore TMS in a systematic and flexible way, miniaturization of TMS for rodent brain studies is a complementary addition to the human studies.
Our investigation is a complementary addition to a previous work (Vörös et al. 2008) in which authors found noisy fluctuations in the magnetosheath as signature of the presence of independent driving sources.
Hence, substituting 1 − y for y and μ − 1 for λμ in (4.27), in view of the complementary addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result follows immediately.
By substituting 1 − y for y and μ − 1 for λμ in (4.16), in view of the complementary addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result follows immediately.
Applying (2.2), (3.5) and (4.21) to (4.20), in view of the Cauchy product and the complementary addition theorem for the Apostol-Bernoulli polynomials, we obtain ∑ m = 0 ∞ ∑ n = 0 ∞ [ ∑ i = 0 n ( n i ) ∑ j = 0 k ( k j ) E i + j ( x ; μ ) E m + n + k − i − j ( y ; λ μ ) ] u m m ! v n n !
Proposition 2.4 Complementary addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials of order α: for a non-negative integer n, B n ( α − x ; λ ) = ( − 1 ) n λ α B n ( x ; λ − 1 ), E n ( α − x ; λ ) = ( − 1 ) n λ α E n ( x ; λ − 1 ).
Applying (2.2) and (3.6) to (4.4), in view of the Cauchy product and the complementary addition theorem for the Apostol-Euler polynomials, we obtain ∑ m = 0 ∞ ∑ n = 0 ∞ [ ∑ i = 0 n ( n i ) ∑ j = 0 k ( k j ) E i + j ( x ; μ ) B m + n + k − i − j ( y ; λ μ ) ] u m m !
Applying (2.1), (3.5) and (3.6) to (3.4), in view of the Cauchy product and the complementary addition theorem of the Apostol-Bernoulli polynomials, we derive ∑ m = 0 ∞ ∑ n = 0 ∞ [ ∑ i = 0 n ( n i ) ∑ j = 0 k ( k j ) B i + j ( x ; μ ) B m + n + k − i − j ( y ; λ μ ) ] u m m ! v n n !
