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Binomial Series01:30

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The binomial series extends the familiar binomial theorem from finite polynomial expansions to infinite series expansions. This distinction is important: the binomial theorem applies to positive integer exponents, while the binomial series applies more broadly, including fractional and negative exponents. It is obtained from the Maclaurin series of (1 + x)m, where m is any real exponent, and the expansion converges for |x| < 1.The familiar binomial theorem...
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Trigonometric identities are equations that relate trigonometric functions and hold for all angles within their domains. A fundamental identity among these is the Pythagorean identity, which arises directly from the geometry of the unit circle. For any angle θ, a point on the unit circle has coordinates (cos⁡ θ, sin ⁡θ), and since the radius of the circle is one, the Pythagorean Theorem gives:This identity serves as the basis for deriving additional identities. Dividing the Pythagorean identity...
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Updated: Jul 17, 2026

Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps
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Published on: October 28, 2018

Transformations and Summations for Bilateral Basic Hypergeometric Series.

Howard S Cohl1, Michael J Schlosser2

  • 1Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, 92694 CA, USA.

Arabian Journal of Mathematics
|July 16, 2026
PubMed
Summary

This study derives new transformation and summation formulas for bilateral basic hypergeometric series. It extends known results, yielding novel quadratic and cubic summations and uncovering new series transformations.

Keywords:
33D1533D45

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Published on: October 28, 2018

Area of Science:

  • Mathematics
  • Special Functions
  • Hypergeometric Series

Background:

  • Bilateral basic hypergeometric series are a complex area of mathematical study.
  • Existing transformations by Bailey are foundational but have room for extension.

Purpose of the Study:

  • To derive novel transformation and summation formulas for bilateral basic hypergeometric series.
  • To explore consequences of bilateral extensions of Bailey's work.
  • To uncover new special cases and summations.

Main Methods:

  • Building upon recent transformations of bilateral basic very-well-poised 8Ψ8 series.
  • Investigating consequences of bilateral extensions of Bailey's transformations.
  • Deriving lower-level transformations using limits and parameter symmetry.

Main Results:

  • New transformation and summation formulas for bilateral basic hypergeometric series.
  • Explicit bilateral quadratic and cubic summations.
  • Novel transformations derived from recent work and parameter symmetry.

Conclusions:

  • The study successfully extends known results in hypergeometric series.
  • New explicit summations and transformations have been discovered.
  • The findings offer new avenues for research in special functions.