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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.27613 |
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| _version_ | 1866911551096619008 |
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| author | Zhou, Jian |
| author_facet | Zhou, Jian |
| contents | We present a large-scale computational study combining arbitrary-precision arithmetic, sequence acceleration, and the PSLQ integer relation algorithm to discover exact closed-form expressions for fundamental constants arising in asymptotic analysis. We compute the Stokes multipliers C_M of the one-dimensional anharmonic oscillators H = p^2/2 + x^2/2 + g x^{2M} for M = 2, 3, ..., 11, extracting 17-30 significant digits from up to 1200 perturbation coefficients computed at 300-digit working precision. The computational pipeline consists of three stages: (i) Rayleigh-Schrodinger recursion in the harmonic oscillator basis, (ii) Richardson extrapolation of order 40-100 to accelerate convergence of ratio sequences, and (iii) PSLQ searches over bases of Gamma-function values and algebraic numbers. This pipeline discovers three new exact identities: C_3^2 pi^4 = 32, C_5^4 Gamma(1/4)^4 pi^5 = 2^{12} 3^2, and C_7^6 Gamma(1/3)^9 pi^6 = 2^{20} 3^3, in addition to confirming the known C_2^2 pi^3 = 6. Equally significant is a negative result: exhaustive PSLQ searches at 30-digit precision with coefficient bounds up to 2000 find no closed form for C_4, strongly suggesting the x^8 case introduces a genuinely new transcendental number. A number-theoretic pattern emerges: closed-form existence correlates with Euler's totient function phi(M-1)/2, which counts algebraically independent Gamma-function transcendentals at denominator M-1. We formulate conjectures connecting computational constant recognition to classical number theory, and provide all code and data for full reproducibility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27613 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-Precision Computation and PSLQ Identification of Stokes Multipliers for Anharmonic Oscillators Zhou, Jian Mathematical Software Numerical Analysis 81Q20, 11Y60, 41A60, 34E05 F.2.1; G.1.2; G.4 We present a large-scale computational study combining arbitrary-precision arithmetic, sequence acceleration, and the PSLQ integer relation algorithm to discover exact closed-form expressions for fundamental constants arising in asymptotic analysis. We compute the Stokes multipliers C_M of the one-dimensional anharmonic oscillators H = p^2/2 + x^2/2 + g x^{2M} for M = 2, 3, ..., 11, extracting 17-30 significant digits from up to 1200 perturbation coefficients computed at 300-digit working precision. The computational pipeline consists of three stages: (i) Rayleigh-Schrodinger recursion in the harmonic oscillator basis, (ii) Richardson extrapolation of order 40-100 to accelerate convergence of ratio sequences, and (iii) PSLQ searches over bases of Gamma-function values and algebraic numbers. This pipeline discovers three new exact identities: C_3^2 pi^4 = 32, C_5^4 Gamma(1/4)^4 pi^5 = 2^{12} 3^2, and C_7^6 Gamma(1/3)^9 pi^6 = 2^{20} 3^3, in addition to confirming the known C_2^2 pi^3 = 6. Equally significant is a negative result: exhaustive PSLQ searches at 30-digit precision with coefficient bounds up to 2000 find no closed form for C_4, strongly suggesting the x^8 case introduces a genuinely new transcendental number. A number-theoretic pattern emerges: closed-form existence correlates with Euler's totient function phi(M-1)/2, which counts algebraically independent Gamma-function transcendentals at denominator M-1. We formulate conjectures connecting computational constant recognition to classical number theory, and provide all code and data for full reproducibility. |
| title | High-Precision Computation and PSLQ Identification of Stokes Multipliers for Anharmonic Oscillators |
| topic | Mathematical Software Numerical Analysis 81Q20, 11Y60, 41A60, 34E05 F.2.1; G.1.2; G.4 |
| url | https://arxiv.org/abs/2603.27613 |