Hiⁿdūš (Old Persian)
|Satrapy of the Achaemenid Empire|
|513 BCE–c. 4th century BCE|
Hiⁿdūš was part of the easternmost territories of the Achaemenid Empire
King of Kings
• 513–499 BCE
|Darius I (first)|
• 358–338 BC
|Historical era||Achaemenid era|
|c. 4th century BCE|
Hindush (Old Persian cuneiform: 𐏃𐎡𐎯𐎢𐏁, Hidūš, also transliterated as Hiⁿdūš since the nasal "n" before consonants was omitted in the Old Persian script, and simplified as Hindūš or sometimes Hindush was a province of the Achaemenid Empire in lower Indus Valley (modern Sindh) established after the Achaemenid conquest circa 500 BC. According to the ancient Greek historian Herodotus, it was the "easternmost province" of the empire. It is believed to have continued as a province until the invasion of the empire by Alexander the Great circa 326 BC.
Hindush was written in Achaemenid inscriptions as Hidūsh (Old Persian cuneiform: 𐏃𐎡𐎯𐎢𐏁, H-i-du-u-š). It is also transliterated as Hiⁿdūš since the nasal "n" before consonants was omitted in the Old Persian script, and simplified as Hindush).
It is widely accepted that the name Hindush derives from Sindhu, the Sanskrit name of the Indus river as well as the region at the lower Indus basin (modern Sindh). The Proto-Iranian sound change *s > h occurred between 850–600 BCE, according to Asko Parpola. The -sh suffix is common among the names of many Achaemenid provinces, such as Harauvatish (the land of Harauvati or Haraxvaiti, i.e., Arachosia) or Bakhtrish (Bactria). Accordingly, Hindush would mean the land of Sindhu.
The Greeks of Asia Minor, who were also part of the Achaemenid empire, called the province 'India'. More precisely, they called the people of the province as 'Indians' ('Ινδοι, Indoi) The loss of the aspirate /h/ was probably due to the dialects of Greek spoken in Asia Minor. Herodotus also generalised the term "Indian" from the people of Hindush to all the people living to the east of Persia, even though he had no knowledge of the geography of the land.
The territory may have corresponded to the area covering the lower and central Indus basin (present day Sindh and the southern Punjab regions of India and Pakistan). To the north of Hindush was Gandāra (spelt as Gaⁿdāra by the Achaememids). These areas remained under Persian control until the invasion by Alexander.
Alternatively, some authors consider that Hindush may have been located in the Punjab area.
According to Herodotus, the 'Indians' participated to the Second Persian invasion of Greece circa 480 BCE. At the final Battle of Platea (479 BCE), they formed one of the main corps of Achaemenid troops (one of "the greatest of the nations"). Indians were still supplying troops and elephants for the Achaemenid army at the Battle of Gaugamela (331 BCE). They are also depicted on the Achaemenid tombs of Naqsh-e Rostam and Persepolis.
Representatives of Hindush are depicted as delegates bringing gifts to the king on the Apadana staircases, and as throne/ dais bearers on the Tripylon and Hall of One Hundred Columns reliefs at Persepolis The representatives of Hindush (as well as Gandara and Thatagus) in each in- stance are characterized by their loincloths, sandals, and exposed upper body, which distinguish them from the representatives of other eastern provinces such as Bactria and Arachosia. 
... that he annexed parts of India as Hindush, the twentieth satrapy of his empire.
Olmstead's Hindush is the Punjāb east of the Indus - as his first Map, “Satrapies of the Persian Empire ”, makes perfectly clear.